{"id":845,"date":"2025-06-20T17:17:07","date_gmt":"2025-06-20T17:17:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=845"},"modified":"2025-08-13T17:30:17","modified_gmt":"2025-08-13T17:30:17","slug":"introduction-to-differential-equations-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-differential-equations-get-stronger\/","title":{"raw":"Introduction to Differential Equations: Get Stronger","rendered":"Introduction to Differential Equations: Get Stronger"},"content":{"raw":"<h2><span data-sheets-root=\"1\">Basics of Differential Equations<\/span><\/h2>\r\n<p class=\"whitespace-normal break-words\"><strong>In the following exercises (1-4), determine the order of the following differential equations.<\/strong><\/p>\r\n\r\n<ol class=\"list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }+y=3{y}^{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^{\\prime\\prime\\prime}+y^{\\prime\\prime}{y}^{\\prime }=3{x}^{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{\\left(\\dfrac{dy}{dt}\\right)}^{2}+8\\dfrac{dy}{dt}+3y=4t[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>For the following exercises (5-9), verify that the following functions are solutions to the given differential equation.<\/strong><\/p>\r\n\r\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"5\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y=2{e}^{\\text{-}x}+x - 1[\/latex] solves [latex]{y}^{\\prime }=x-y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y=\\dfrac{1}{1-x}[\/latex] solves [latex]{y}^{\\prime }={y}^{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y=4+\\text{ln}x[\/latex] solves [latex]x{y}^{\\prime }=1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y=2{e}^{x}-x - 1[\/latex] solves [latex]{y}^{\\prime }=y+x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y=\\pi {e}^{\\text{-}\\cos{x}}[\/latex] solves [latex]{y}^{\\prime }=y\\sin{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Verify the following general solutions (10-14) and find the particular solution.<\/strong><\/p>\r\n\r\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"10\">\r\n \t<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{3}[\/latex] that passes through [latex]\\left(1,4.75\\right)[\/latex], given that [latex]y=C+\\dfrac{3{x}^{4}}{4}[\/latex] is a general solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=2xy[\/latex] that passes through [latex]\\left(0,\\dfrac{1}{2}\\right)[\/latex], given that [latex]y=C{e}^{{x}^{2}}[\/latex] is a general solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }{x}^{2}=y[\/latex] that passes through [latex]\\left(1,\\dfrac{2}{e}\\right)[\/latex], given that [latex]y=C{e}^{\\dfrac{\\text{-}1}{x}}[\/latex] is a general solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]\\dfrac{du}{dt}=\\tan{u}[\/latex] that passes through [latex]\\left(1,\\dfrac{\\pi }{2}\\right)[\/latex], given that [latex]u={\\sin}^{-1}\\left({e}^{C+t}\\right)[\/latex] is a general solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }\\left(1-{x}^{2}\\right)=1+y[\/latex] that passes through [latex]\\left(0,-2\\right)[\/latex], given that [latex]y=C\\dfrac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex] is a general solution.<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>For the following problems (15-19), find the general solution to the differential equation.<\/strong><\/p>\r\n\r\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"15\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=\\text{ln}x+\\tan{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }={4}^{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=2t\\sqrt{{t}^{2}+16}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x}^{\\prime }=t\\sqrt{4+t}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=\\dfrac{y}{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Solve the following initial-value problems (20-21) starting from [latex]y\\left(t=0\\right)=1[\/latex] and [latex]y\\left(t=0\\right)=-1[\/latex]. Draw both solutions on the same graph.<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"20\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=\\text{-}t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=\\text{-}y[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Solve the following initial-value problems (22-24) starting from [latex]{y}_{0}=10[\/latex]. At what time does [latex]y[\/latex] increase to [latex]100[\/latex] or drop to [latex]1?[\/latex]<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"22\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=4t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=-2y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}={e}^{-4t}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from [latex]y\\left(t=0\\right)=-10[\/latex] to [latex]y\\left(t=0\\right)=10[\/latex] increasing by [latex]2[\/latex]. Is there some critical point where the behavior of the solution begins to change?<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"25\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x{y}^{\\prime }=y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=x+y[\/latex] (Hint: [latex]y=C{e}^{x}-x - 1[\/latex] is the general solution)<\/li>\r\n<\/ol>\r\n<strong>For the following exercises (27-31), solve each problem.<\/strong>\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"27\">\r\n \t<li class=\"whitespace-normal break-words\">Find the general solution to describe the velocity of a ball of mass [latex]1\\text{lb}[\/latex] that is thrown upward at a rate [latex]a[\/latex] ft\/sec.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">You throw two objects with differing masses [latex]{m}<em>{1}[\/latex] and [latex]{m}<\/em>{2}[\/latex] upward into the air with the same initial velocity [latex]a[\/latex] ft\/s. What is the difference in their velocity after [latex]1[\/latex] second?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]y=B{e}^{3t}[\/latex] into [latex]{y}^{\\prime }-y=8{e}^{3t}[\/latex] to find a particular solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]y=a+bt+c{t}^{2}[\/latex] into [latex]{y}^{\\prime }+y=1+{t}^{2}[\/latex] to find a particular solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve [latex]{y}^{\\prime }={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex] and solve [latex]{y}^{\\prime }=1[\/latex] with the same initial condition. As [latex]k[\/latex] approaches [latex]0[\/latex], what do you notice?<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Direction Fields and Euler's Method<\/span><\/h2>\r\n<p dir=\"auto\"><strong>For the following problems (1-2), use the direction field below from the differential equation [latex]y^{\\prime} =-2y[\/latex]. Sketch the graph of the solution for the given initial conditions.<\/strong><\/p>\r\n<p dir=\"auto\"><span id=\"fs-id1170571153152\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right at 0. The arrows above the x-axis point down and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are. Likewise, the arrows below the x-axis point up and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are.\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234125\/CNX_Calc_Figure_08_02_201.jpg\" alt=\"A direction field with horizontal arrows pointing to the right at 0. The arrows above the x-axis point down and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are. Likewise, the arrows below the x-axis point up and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\r\n\r\n<ol dir=\"auto\">\r\n \t<li>[latex]y\\left(0\\right)=0[\/latex]<\/li>\r\n \t<li>Are there any equilibria? What are their stabilities?<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following problems (3-4), use the direction field below from the differential equation [latex]y^{\\prime} ={y}^{2}-2y[\/latex]. Sketch the graph of the solution for the given initial conditions.<\/strong><\/p>\r\n<p dir=\"auto\"><span id=\"fs-id1170571240273\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y &gt; 2 and for y &lt; 0. The arrows point down for 0 &lt; y &lt; 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are.\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234129\/CNX_Calc_Figure_08_02_205.jpg\" alt=\"A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y &gt; 2 and for y &lt; 0. The arrows point down for 0 &lt; y &lt; 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\r\n\r\n<ol dir=\"auto\" start=\"3\">\r\n \t<li>[latex]y\\left(0\\right)=1[\/latex]<\/li>\r\n \t<li>Are there any equilibria? What are their stabilities?<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>Draw the direction field for the following differential equations (5-6), then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field?<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"5\">\r\n \t<li>[latex]y^{\\prime} ={e}^{t}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dy}{dt}=t{e}^{t}[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>Draw the directional field for the following differential equations (7-9). What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"7\">\r\n \t<li>[latex]y^{\\prime} ={y}^{2}-1[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =1-{y}^{2}-{x}^{2}[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =3y+xy[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>Match the direction field with the given differential equations (10-11). Explain your selections.<\/strong><\/p>\r\n<p dir=\"auto\"><span id=\"fs-id1170571423084\" data-type=\"media\" data-alt=\"A direction field with arrows pointing down and to the right in quadrants two and three. After crossing the y axis, the arrows change direction and point up to the right.\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234146\/CNX_Calc_Figure_08_02_219a.jpg\" alt=\"A direction field with arrows pointing down and to the right in quadrants two and three. After crossing the y axis, the arrows change direction and point up to the right.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423097\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the left in quadrants two and three. In crossing the y axis, the arrows switch and point upward in quadrants one and four.\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234149\/CNX_Calc_Figure_08_02_219b.jpg\" alt=\"A direction field with horizontal arrows pointing to the left in quadrants two and three. In crossing the y axis, the arrows switch and point upward in quadrants one and four.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423110\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x axis. Above, the arrows point down and to the right, and below, the arrows point up and to the right. The further from the x axis, the more vertical the arrows become.\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234152\/CNX_Calc_Figure_08_02_219c.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x axis. Above, the arrows point down and to the right, and below, the arrows point up and to the right. The further from the x axis, the more vertical the arrows become.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423123\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows on the x and y axes. The arrows point down and to the right in quadrants one and three. They point up and to the right in quadrants two and four.\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234154\/CNX_Calc_Figure_08_02_219d.jpg\" alt=\"A direction field with horizontal arrows on the x and y axes. The arrows point down and to the right in quadrants one and three. They point up and to the right in quadrants two and four.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571306925\" data-type=\"media\" data-alt=\"A direction field with arrows pointing up in quadrants two and three, to the right on the y axis, and down in quadrants one and four.\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234157\/CNX_Calc_Figure_08_02_219e.jpg\" alt=\"A direction field with arrows pointing up in quadrants two and three, to the right on the y axis, and down in quadrants one and four.\" width=\"322\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\r\n\r\n<ol dir=\"auto\" start=\"10\">\r\n \t<li>[latex]y^{\\prime} =-3t[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =\\dfrac{1}{2}y+t[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Match the direction field with the given differential equations (12-14). Explain your selections.<\/strong>\r\n\r\n<span id=\"fs-id1170571469184\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up, and in quadrants two and four, they point down.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234200\/CNX_Calc_Figure_08_02_220a.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up, and in quadrants two and four, they point down.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571469197\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up and to the right, and in quadrants two and four, the arrows point down and to the right.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234202\/CNX_Calc_Figure_08_02_220b.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up and to the right, and in quadrants two and four, the arrows point down and to the right.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118906\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants two and three, the arrows point down, and in quadrants one and four, the arrows point up.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234205\/CNX_Calc_Figure_08_02_220c.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants two and three, the arrows point down, and in quadrants one and four, the arrows point up.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118919\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x axis. The arrows point up and to the right in all quadrants. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they are.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234208\/CNX_Calc_Figure_08_02_220d.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x axis. The arrows point up and to the right in all quadrants. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they are.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118937\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows on the y axis. The arrows are also more horizontal closer to y = 1.5, y = -1.5, and the y axis. For y &gt; 1.5 and x &lt; 0, for y &lt; -1.5 and x &lt; 0, and for -1.5 &lt; y &lt; 1.5 and x &gt; 0-, the arrows point down. For y&gt; 1.5 and x &gt; 0, for y &lt; -1.5, for y &lt; -1.5 and x &gt; 0, and for -1.5 &lt; y &lt; 1.5 and x &lt; 0, the arrows point up.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234211\/CNX_Calc_Figure_08_02_220e.jpg\" alt=\"A direction field with horizontal arrows on the y axis. The arrows are also more horizontal closer to y = 1.5, y = -1.5, and the y axis. For y &gt; 1.5 and x &lt; 0, for y &lt; -1.5 and x &lt; 0, and for -1.5 &lt; y &lt; 1.5 and x &gt; 0-, the arrows point down. For y&gt; 1.5 and x &gt; 0, for y &lt; -1.5, for y &lt; -1.5 and x &gt; 0, and for -1.5 &lt; y &lt; 1.5 and x &lt; 0, the arrows point up.\" width=\"323\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<ol dir=\"auto\" start=\"12\">\r\n \t<li>[latex]y^{\\prime} =t\\sin{y}[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =t\\tan{y}[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} ={y}^{2}{t}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>Estimate the following solutions (15-19) using Euler\u2019s method with [latex]n=5[\/latex] steps over the interval [latex]t=\\left[0,1\\right][\/latex]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler\u2019s method. How accurate is Euler\u2019s method?<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"15\">\r\n \t<li>[latex]y^{\\prime} ={t}^{2}[\/latex]<\/li>\r\n \t<li>[latex]{y}^{\\prime }=y+{t}^{2},y\\left(0\\right)=3[\/latex]. Exact solution is [latex]y=5{e}^{t}-2-{t}^{2}-2t[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} ={e}^{\\left(x+y\\right)},y\\left(0\\right)=-1[\/latex]. Exact solution is [latex]y=\\text{-}\\text{ln}\\left(e+1-{e}^{x}\\right)[\/latex]<\/li>\r\n \t<li>[latex]{y}^{\\prime }={2}^{x},y\\left(0\\right)=0[\/latex]. Exact solution is [latex]y=\\dfrac{{2}^{x}-1}{\\text{ln}\\left(2\\right)}[\/latex]<\/li>\r\n \t<li>[latex]{y}^{\\prime }=-5t,y\\left(0\\right)=-2[\/latex]. Exact solution is [latex]y=-\\dfrac{5}{2}{t}^{2}-2[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following exercises (20-21), consider the initial-value problem [latex]y^{\\prime} =-2y,y\\left(0\\right)=2[\/latex].<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"20\">\r\n \t<li>Draw the directional field of this differential equation.<\/li>\r\n \t<li>By calculator or computer, approximate the solution using Euler\u2019s Method at [latex]t=10[\/latex] using [latex]h=100[\/latex].<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Separation of Variables<\/span><\/h2>\r\n<p dir=\"auto\"><strong>For the following exercises (1-2), solve the following initial-value problems with the initial condition [latex]{y}_{0}=0[\/latex] and graph the solution.<\/strong><\/p>\r\n\r\n<ol dir=\"auto\">\r\n \t<li>[latex]\\dfrac{dy}{dt}=y+1[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dy}{dt}=\\text{-}y - 1[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following exercises (3-7), find the general solution to the differential equation.<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"3\">\r\n \t<li>[latex]{x}^{2}y^{\\prime} =\\left(x+1\\right)y[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =2x{y}^{2}[\/latex]<\/li>\r\n \t<li>[latex]2x\\dfrac{dy}{dx}={y}^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(1+x\\right)y^{\\prime} =\\left(x+2\\right)\\left(y - 1\\right)[\/latex]<\/li>\r\n \t<li>[latex]t\\dfrac{dy}{dt}=\\sqrt{1-{y}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following exercises (8-12), find the solution to the initial-value problem.<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"8\">\r\n \t<li>[latex]y^{\\prime} ={e}^{y-x},y\\left(0\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dy}{dx}={y}^{3}x{e}^{{x}^{2}},y\\left(0\\right)=1[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =\\dfrac{x}{{\\text{sech}}^{2}y},y\\left(0\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dx}{dt}=\\text{ln}\\left(t\\right)\\sqrt{1-{x}^{2}},x\\left(1\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} ={e}^{y}{5}^{x},y\\left(0\\right)=\\text{ln}\\left(\\text{ln}\\left(5\\right)\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following problems (13-15), use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"13\">\r\n \t<li>[latex]y^{\\prime} =1 - 2y[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} ={y}^{3}{e}^{x}[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =y\\text{ln}\\left(x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<strong>For the following exercises (16-17), solve each problem.<\/strong>\r\n<ol dir=\"auto\" start=\"16\">\r\n \t<li>A drug is administered intravenously to a patient at a rate [latex]r[\/latex] mg\/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, [latex]d[\/latex]. Set up and solve the differential equation, assuming there is no drug initially present in the body.<\/li>\r\n \t<li>A tank contains [latex]1[\/latex] kilogram of salt dissolved in [latex]100[\/latex] liters of water. A salt solution of [latex]0.1[\/latex] kg salt\/L is pumped into the tank at a rate of [latex]2[\/latex] L\/min and is drained at the same rate. Solve for the salt concentration at time [latex]t[\/latex]. Assume the tank is well mixed.<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following problems (20-22), use Newton\u2019s law of cooling.<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"20\">\r\n \t<li>The liquid base of an ice cream has an initial temperature of [latex]210^\\circ\\text{F}[\/latex] before it is placed in a freezer with a constant temperature of [latex]20^\\circ\\text{F}\\text{.}[\/latex] After [latex]2[\/latex] hours, the temperature of the ice-cream base has decreased to [latex]170^\\circ\\text{F}\\text{.}[\/latex] At what time will the ice cream be ready to eat? (Assume [latex]30^\\circ\\text{F}[\/latex] is the optimal eating temperature.)<\/li>\r\n \t<li>You have a cup of coffee at temperature [latex]70^\\circ\\text{C}[\/latex] and the ambient temperature in the room is [latex]20^\\circ\\text{C}\\text{.}[\/latex] Assuming a cooling rate [latex]k\\text{ of }0.125[\/latex], write and solve the differential equation to describe the temperature of the coffee with respect to time.<\/li>\r\n \t<li>You have a cup of coffee at temperature [latex]70^\\circ\\text{C}[\/latex] and you immediately pour in [latex]1[\/latex] part milk to [latex]5[\/latex] parts coffee. The milk is initially at temperature [latex]1^\\circ\\text{C}\\text{.}[\/latex] Write and solve the differential equation that governs the temperature of this coffee.<\/li>\r\n<\/ol>\r\n<p dir=\"auto\"><strong>For the following exercises (23-25), solve each problem.<\/strong><\/p>\r\n\r\n<ol dir=\"auto\" start=\"23\">\r\n \t<li>Solve the generic problem [latex]y^{\\prime} =ay+b[\/latex] with initial condition [latex]y\\left(0\\right)=c[\/latex].<\/li>\r\n \t<li>Assume an initial nutrient amount of [latex]I[\/latex] kilograms in a tank with [latex]L[\/latex] liters. Assume a concentration of [latex]c[\/latex] kg\/L being pumped in at a rate of [latex]r[\/latex] L\/min. The tank is well mixed and is drained at a rate of [latex]r[\/latex] L\/min. Find the equation describing the amount of nutrient in the tank.<\/li>\r\n \t<li>Leaves accumulate on the forest floor at a rate of [latex]4[\/latex] g\/cm\u00b2\/yr. These leaves decompose at a rate of [latex]10\\text{%}[\/latex] per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">First-Order Linear Equations and Applications<\/span><\/h2>\r\n<p dir=\"ltr\" data-pm-slice=\"1 3 []\"><strong>Are the following differential equations linear (1-2)? Explain your reasoning.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" data-tight=\"true\">\r\n \t<li>[latex]\\dfrac{dy}{dt}=ty[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} ={x}^{3}+{e}^{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Write the following first-order differential equations in standard form (3-5).<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"3\" data-tight=\"true\">\r\n \t<li>[latex]y^{\\prime} ={x}^{3}y+\\sin{x}[\/latex]<\/li>\r\n \t<li>[latex]\\text{-}xy^{\\prime} =\\left(3x+2\\right)y+x{e}^{x}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dy}{dt}=yx\\left(x+1\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>What are the integrating factors for the following differential equations (6-7)?<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"6\" data-tight=\"true\">\r\n \t<li>[latex]y^{\\prime} +{e}^{x}y=\\sin{x}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{dy}{dx}=\\text{tanh}\\left(x\\right)y+1[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Solve the following differential equations (8-12) by using integrating factors.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"8\" data-tight=\"true\">\r\n \t<li>[latex]y^{\\prime} =3y+2[\/latex]<\/li>\r\n \t<li>[latex]xy^{\\prime} =3y - 6{x}^{2}[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime} =3x+xy[\/latex]<\/li>\r\n \t<li>[latex]\\sin\\left(x\\right)y^{\\prime} =y+2x[\/latex]<\/li>\r\n \t<li>[latex]xy^{\\prime} =3y+{x}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Solve the following differential equations (13-16). Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"13\" data-tight=\"true\">\r\n \t<li>[latex]\\left(x+2\\right)y^{\\prime} =2y - 1[\/latex]<\/li>\r\n \t<li>[latex]xy^{\\prime} +\\dfrac{y}{2}=\\sin\\left(3t\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+1\\right)y^{\\prime} =3y+{x}^{2}+2x+1[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{{x}^{2}+1}y^{\\prime} =y+2[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Solve the following initial-value problems (17-21) by using integrating factors.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"17\" data-tight=\"true\">\r\n \t<li>[latex]y^{\\prime} +y=x,y\\left(0\\right)=3[\/latex]<\/li>\r\n \t<li>[latex]xy^{\\prime} =y - 3{x}^{3},y\\left(1\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]\\left(1+{x}^{2}\\right)y^{\\prime} =y - 1,y\\left(0\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]\\left(2+x\\right)y^{\\prime} =y+2+x,y\\left(0\\right)=0[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{x}y^{\\prime} =y+2x,y\\left(0\\right)=1[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>For the following exercises (22-24), solve each problem.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"22\" data-tight=\"true\">\r\n \t<li>A falling object of mass [latex]m[\/latex] can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity given an initial velocity of [latex]0[\/latex].<\/li>\r\n \t<li>Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)<\/li>\r\n \t<li>Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8[\/latex] m\/s\u00b2 and the proportionality constant is [latex]4?[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>For the following problems (25-26), determine how parameter [latex]a[\/latex] affects the solution.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"25\" data-tight=\"true\">\r\n \t<li>Solve the generic equation [latex]y^{\\prime} =ax+y[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/li>\r\n \t<li>Solve the generic equation [latex]y^{\\prime} =ax+xy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\" data-pm-slice=\"1 3 []\"><strong>For the following problem, consider the logistic equation in the form [latex]P\\prime =CP-{P}^{2}[\/latex]. Find the stability of the equilibria.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"27\" data-tight=\"true\">\r\n \t<li>\r\n<p dir=\"ltr\">[latex]C=0[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Solve the logistic equation for the given conditions.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"28\" data-tight=\"true\">\r\n \t<li>\r\n<p dir=\"ltr\">Solve the logistic equation for [latex]C=10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>For the following exercises (29-34), solve each problem.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"29\" data-tight=\"true\">\r\n \t<li>A population of deer inside a park has a carrying capacity of [latex]200[\/latex] and a growth rate of [latex]2\\text{%}[\/latex]. If the initial population is [latex]50[\/latex] deer, what is the population of deer at any given time?<\/li>\r\n \t<li>Bacteria grow at a rate of [latex]20\\text{%}[\/latex] per hour in a petri dish. If there is initially one bacterium and a carrying capacity of [latex]1[\/latex] million cells, how long does it take to reach [latex]500,000[\/latex] cells?<\/li>\r\n \t<li>Two monkeys are placed on an island. After [latex]5[\/latex] years, there are [latex]8[\/latex] monkeys, and the estimated carrying capacity is [latex]25[\/latex] monkeys. When does the population of monkeys reach [latex]16[\/latex] monkeys?<\/li>\r\n \t<li>It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant [latex]k[\/latex], as [latex]P\\prime =0.4P\\left(1-\\dfrac{P}{10000}\\right)-kP[\/latex]. Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]2000[\/latex] fish.<\/li>\r\n \t<li>Lets add in a minimal threshold value for the species to survive, [latex]T[\/latex], which changes the differential equation to [latex]P\\prime \\left(t\\right)=rP\\left(1-\\dfrac{P}{K}\\right)\\left(1-\\dfrac{T}{P}\\right)[\/latex].\u00a0Bengal tigers in a conservation park have a carrying capacity of [latex]100[\/latex] and need a minimum of [latex]10[\/latex] to survive. If they grow in population at a rate of [latex]1\\text{%}[\/latex] per year, with an initial population of [latex]15[\/latex] tigers, solve for the number of tigers present.<\/li>\r\n \t<li>The population of mountain lions in Northern Arizona has an estimated carrying capacity of [latex]250[\/latex] and grows at a rate of [latex]0.25\\text{%}[\/latex] per year and there must be [latex]25[\/latex] for the population to survive. With an initial population of [latex]30[\/latex] mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least [latex]100[\/latex]?)**<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>The following questions (35-37) consider the <span class=\"no-emphasis\" data-type=\"term\">Gompertz equation<\/span>, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.<\/strong><\/p>\r\n\r\n<ol class=\"tight\" dir=\"ltr\" start=\"35\" data-tight=\"true\">\r\n \t<li>The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day [latex]1[\/latex] and assuming [latex]\\alpha =0.1[\/latex] and a carrying capacity of [latex]10[\/latex] million cells, how long does it take to reach \"detection\" stage at [latex]5[\/latex] million cells?<\/li>\r\n \t<li>It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached [latex]7[\/latex] billion. Was logistic growth or Gompertz growth more accurate, considering world population reached [latex]7[\/latex] billion on October [latex]31,2011?[\/latex]<\/li>\r\n \t<li>When does population increase the fastest in the threshold logistic equation [latex]P\\prime \\left(t\\right)=rP\\left(1-\\dfrac{P}{K}\\right)\\left(1-\\dfrac{T}{P}\\right)?[\/latex]<\/li>\r\n<\/ol>\r\n<p dir=\"ltr\"><strong>Below is a table of the populations of whooping cranes in the wild from [latex]1940\\text{ to }2000[\/latex]. The population rebounded from near extinction after conservation efforts began. The following problems (38-41) consider applying population models to fit the data. Assume a carrying capacity of [latex]10,000[\/latex] cranes. Fit the data assuming years since [latex]1940[\/latex] (so your initial population at time [latex]0[\/latex] would be [latex]22[\/latex] cranes).<\/strong><\/p>\r\n\r\n<table id=\"fs-id1170571715330\" class=\"unnumbered\" summary=\"A table with eight rows and two columns. The first column has the label \" data-label=\"\"><caption><em data-effect=\"italics\">Source:<\/em> https:\/\/www.savingcranes.org\/images\/stories\/site_images\/conservation\/whooping_crane\/pdfs\/historic_wc_numbers.pdf<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">Year (years since conservation began)<\/th>\r\n<th data-align=\"left\">Whooping Crane Population<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1940\\left(0\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]22[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1950\\left(10\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]31[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1960\\left(20\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]36[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1970\\left(30\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]57[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1980\\left(40\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]91[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1990\\left(50\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]159[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2000\\left(60\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]256[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol dir=\"ltr\" start=\"38\">\r\n \t<li>Find the equation and parameter [latex]r[\/latex] that best fit the data for the logistic equation.<\/li>\r\n \t<li>Find the equation and parameters [latex]r[\/latex] and [latex]T[\/latex] that best fit the data for the threshold logistic equation.<\/li>\r\n \t<li>Find the equation and parameter [latex]\\alpha[\/latex] that best fit the data for the Gompertz equation.<\/li>\r\n \t<li>Using the three equations found in the previous problems, estimate the population in [latex]2010[\/latex] (year [latex]70[\/latex] after conservation). The real population measured at that time was [latex]437[\/latex]. Which model is most accurate?<\/li>\r\n<\/ol>","rendered":"<h2><span data-sheets-root=\"1\">Basics of Differential Equations<\/span><\/h2>\n<p class=\"whitespace-normal break-words\"><strong>In the following exercises (1-4), determine the order of the following differential equations.<\/strong><\/p>\n<ol class=\"list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }+y=3{y}^{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y^{\\prime\\prime\\prime}+y^{\\prime\\prime}{y}^{\\prime }=3{x}^{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{\\left(\\dfrac{dy}{dt}\\right)}^{2}+8\\dfrac{dy}{dt}+3y=4t[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>For the following exercises (5-9), verify that the following functions are solutions to the given differential equation.<\/strong><\/p>\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"5\">\n<li class=\"whitespace-normal break-words\">[latex]y=2{e}^{\\text{-}x}+x - 1[\/latex] solves [latex]{y}^{\\prime }=x-y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y=\\dfrac{1}{1-x}[\/latex] solves [latex]{y}^{\\prime }={y}^{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y=4+\\text{ln}x[\/latex] solves [latex]x{y}^{\\prime }=1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y=2{e}^{x}-x - 1[\/latex] solves [latex]{y}^{\\prime }=y+x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y=\\pi {e}^{\\text{-}\\cos{x}}[\/latex] solves [latex]{y}^{\\prime }=y\\sin{x}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Verify the following general solutions (10-14) and find the particular solution.<\/strong><\/p>\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"10\">\n<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{3}[\/latex] that passes through [latex]\\left(1,4.75\\right)[\/latex], given that [latex]y=C+\\dfrac{3{x}^{4}}{4}[\/latex] is a general solution.<\/li>\n<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=2xy[\/latex] that passes through [latex]\\left(0,\\dfrac{1}{2}\\right)[\/latex], given that [latex]y=C{e}^{{x}^{2}}[\/latex] is a general solution.<\/li>\n<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }{x}^{2}=y[\/latex] that passes through [latex]\\left(1,\\dfrac{2}{e}\\right)[\/latex], given that [latex]y=C{e}^{\\dfrac{\\text{-}1}{x}}[\/latex] is a general solution.<\/li>\n<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]\\dfrac{du}{dt}=\\tan{u}[\/latex] that passes through [latex]\\left(1,\\dfrac{\\pi }{2}\\right)[\/latex], given that [latex]u={\\sin}^{-1}\\left({e}^{C+t}\\right)[\/latex] is a general solution.<\/li>\n<li class=\"whitespace-normal break-words\">Find the particular solution to the differential equation [latex]{y}^{\\prime }\\left(1-{x}^{2}\\right)=1+y[\/latex] that passes through [latex]\\left(0,-2\\right)[\/latex], given that [latex]y=C\\dfrac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex] is a general solution.<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>For the following problems (15-19), find the general solution to the differential equation.<\/strong><\/p>\n<ol class=\"list-decimal space-y-1.5 pl-7\" start=\"15\">\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=\\text{ln}x+\\tan{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }={4}^{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=2t\\sqrt{{t}^{2}+16}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{x}^{\\prime }=t\\sqrt{4+t}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=\\dfrac{y}{x}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Solve the following initial-value problems (20-21) starting from [latex]y\\left(t=0\\right)=1[\/latex] and [latex]y\\left(t=0\\right)=-1[\/latex]. Draw both solutions on the same graph.<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"20\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=\\text{-}t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=\\text{-}y[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Solve the following initial-value problems (22-24) starting from [latex]{y}_{0}=10[\/latex]. At what time does [latex]y[\/latex] increase to [latex]100[\/latex] or drop to [latex]1?[\/latex]<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"22\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=4t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}=-2y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{dy}{dt}={e}^{-4t}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from [latex]y\\left(t=0\\right)=-10[\/latex] to [latex]y\\left(t=0\\right)=10[\/latex] increasing by [latex]2[\/latex]. Is there some critical point where the behavior of the solution begins to change?<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"25\">\n<li class=\"whitespace-normal break-words\">[latex]x{y}^{\\prime }=y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{y}^{\\prime }=x+y[\/latex] (Hint: [latex]y=C{e}^{x}-x - 1[\/latex] is the general solution)<\/li>\n<\/ol>\n<p><strong>For the following exercises (27-31), solve each problem.<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\" start=\"27\">\n<li class=\"whitespace-normal break-words\">Find the general solution to describe the velocity of a ball of mass [latex]1\\text{lb}[\/latex] that is thrown upward at a rate [latex]a[\/latex] ft\/sec.<\/li>\n<li class=\"whitespace-normal break-words\">You throw two objects with differing masses [latex]{m}<em>{1}[\/latex] and [latex]{m}<\/em>{2}[\/latex] upward into the air with the same initial velocity [latex]a[\/latex] ft\/s. What is the difference in their velocity after [latex]1[\/latex] second?<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]y=B{e}^{3t}[\/latex] into [latex]{y}^{\\prime }-y=8{e}^{3t}[\/latex] to find a particular solution.<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]y=a+bt+c{t}^{2}[\/latex] into [latex]{y}^{\\prime }+y=1+{t}^{2}[\/latex] to find a particular solution.<\/li>\n<li class=\"whitespace-normal break-words\">Solve [latex]{y}^{\\prime }={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex] and solve [latex]{y}^{\\prime }=1[\/latex] with the same initial condition. As [latex]k[\/latex] approaches [latex]0[\/latex], what do you notice?<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Direction Fields and Euler&#8217;s Method<\/span><\/h2>\n<p dir=\"auto\"><strong>For the following problems (1-2), use the direction field below from the differential equation [latex]y^{\\prime} =-2y[\/latex]. Sketch the graph of the solution for the given initial conditions.<\/strong><\/p>\n<p dir=\"auto\"><span id=\"fs-id1170571153152\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right at 0. The arrows above the x-axis point down and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are. Likewise, the arrows below the x-axis point up and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234125\/CNX_Calc_Figure_08_02_201.jpg\" alt=\"A direction field with horizontal arrows pointing to the right at 0. The arrows above the x-axis point down and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are. Likewise, the arrows below the x-axis point up and to the right. The further away from the x-axis, the steeper the arrows are, and the closer to the x-axis, the flatter the arrows are.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<ol dir=\"auto\">\n<li>[latex]y\\left(0\\right)=0[\/latex]<\/li>\n<li>Are there any equilibria? What are their stabilities?<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following problems (3-4), use the direction field below from the differential equation [latex]y^{\\prime} ={y}^{2}-2y[\/latex]. Sketch the graph of the solution for the given initial conditions.<\/strong><\/p>\n<p dir=\"auto\"><span id=\"fs-id1170571240273\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y &gt; 2 and for y &lt; 0. The arrows point down for 0 &lt; y &lt; 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234129\/CNX_Calc_Figure_08_02_205.jpg\" alt=\"A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y &gt; 2 and for y &lt; 0. The arrows point down for 0 &lt; y &lt; 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<ol dir=\"auto\" start=\"3\">\n<li>[latex]y\\left(0\\right)=1[\/latex]<\/li>\n<li>Are there any equilibria? What are their stabilities?<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>Draw the direction field for the following differential equations (5-6), then solve the differential equation. Draw your solution on top of the direction field. Does your solution follow along the arrows on your direction field?<\/strong><\/p>\n<ol dir=\"auto\" start=\"5\">\n<li>[latex]y^{\\prime} ={e}^{t}[\/latex]<\/li>\n<li>[latex]\\dfrac{dy}{dt}=t{e}^{t}[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>Draw the directional field for the following differential equations (7-9). What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?<\/strong><\/p>\n<ol dir=\"auto\" start=\"7\">\n<li>[latex]y^{\\prime} ={y}^{2}-1[\/latex]<\/li>\n<li>[latex]y^{\\prime} =1-{y}^{2}-{x}^{2}[\/latex]<\/li>\n<li>[latex]y^{\\prime} =3y+xy[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>Match the direction field with the given differential equations (10-11). Explain your selections.<\/strong><\/p>\n<p dir=\"auto\"><span id=\"fs-id1170571423084\" data-type=\"media\" data-alt=\"A direction field with arrows pointing down and to the right in quadrants two and three. After crossing the y axis, the arrows change direction and point up to the right.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234146\/CNX_Calc_Figure_08_02_219a.jpg\" alt=\"A direction field with arrows pointing down and to the right in quadrants two and three. After crossing the y axis, the arrows change direction and point up to the right.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423097\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the left in quadrants two and three. In crossing the y axis, the arrows switch and point upward in quadrants one and four.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234149\/CNX_Calc_Figure_08_02_219b.jpg\" alt=\"A direction field with horizontal arrows pointing to the left in quadrants two and three. In crossing the y axis, the arrows switch and point upward in quadrants one and four.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423110\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x axis. Above, the arrows point down and to the right, and below, the arrows point up and to the right. The further from the x axis, the more vertical the arrows become.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234152\/CNX_Calc_Figure_08_02_219c.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x axis. Above, the arrows point down and to the right, and below, the arrows point up and to the right. The further from the x axis, the more vertical the arrows become.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571423123\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows on the x and y axes. The arrows point down and to the right in quadrants one and three. They point up and to the right in quadrants two and four.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234154\/CNX_Calc_Figure_08_02_219d.jpg\" alt=\"A direction field with horizontal arrows on the x and y axes. The arrows point down and to the right in quadrants one and three. They point up and to the right in quadrants two and four.\" width=\"312\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571306925\" data-type=\"media\" data-alt=\"A direction field with arrows pointing up in quadrants two and three, to the right on the y axis, and down in quadrants one and four.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234157\/CNX_Calc_Figure_08_02_219e.jpg\" alt=\"A direction field with arrows pointing up in quadrants two and three, to the right on the y axis, and down in quadrants one and four.\" width=\"322\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<ol dir=\"auto\" start=\"10\">\n<li>[latex]y^{\\prime} =-3t[\/latex]<\/li>\n<li>[latex]y^{\\prime} =\\dfrac{1}{2}y+t[\/latex]<\/li>\n<\/ol>\n<p><strong>Match the direction field with the given differential equations (12-14). Explain your selections.<\/strong><\/p>\n<p><span id=\"fs-id1170571469184\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up, and in quadrants two and four, they point down.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234200\/CNX_Calc_Figure_08_02_220a.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up, and in quadrants two and four, they point down.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571469197\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up and to the right, and in quadrants two and four, the arrows point down and to the right.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234202\/CNX_Calc_Figure_08_02_220b.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up and to the right, and in quadrants two and four, the arrows point down and to the right.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118906\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants two and three, the arrows point down, and in quadrants one and four, the arrows point up.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234205\/CNX_Calc_Figure_08_02_220c.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants two and three, the arrows point down, and in quadrants one and four, the arrows point up.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118919\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows pointing to the right on the x axis. The arrows point up and to the right in all quadrants. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they are.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234208\/CNX_Calc_Figure_08_02_220d.jpg\" alt=\"A direction field with horizontal arrows pointing to the right on the x axis. The arrows point up and to the right in all quadrants. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they are.\" width=\"310\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170571118937\" data-type=\"media\" data-alt=\"A direction field with horizontal arrows on the y axis. The arrows are also more horizontal closer to y = 1.5, y = -1.5, and the y axis. For y &gt; 1.5 and x &lt; 0, for y &lt; -1.5 and x &lt; 0, and for -1.5 &lt; y &lt; 1.5 and x &gt; 0-, the arrows point down. For y&gt; 1.5 and x &gt; 0, for y &lt; -1.5, for y &lt; -1.5 and x &gt; 0, and for -1.5 &lt; y &lt; 1.5 and x &lt; 0, the arrows point up.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234211\/CNX_Calc_Figure_08_02_220e.jpg\" alt=\"A direction field with horizontal arrows on the y axis. The arrows are also more horizontal closer to y = 1.5, y = -1.5, and the y axis. For y &gt; 1.5 and x &lt; 0, for y &lt; -1.5 and x &lt; 0, and for -1.5 &lt; y &lt; 1.5 and x &gt; 0-, the arrows point down. For y&gt; 1.5 and x &gt; 0, for y &lt; -1.5, for y &lt; -1.5 and x &gt; 0, and for -1.5 &lt; y &lt; 1.5 and x &lt; 0, the arrows point up.\" width=\"323\" height=\"303\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<ol dir=\"auto\" start=\"12\">\n<li>[latex]y^{\\prime} =t\\sin{y}[\/latex]<\/li>\n<li>[latex]y^{\\prime} =t\\tan{y}[\/latex]<\/li>\n<li>[latex]y^{\\prime} ={y}^{2}{t}^{3}[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>Estimate the following solutions (15-19) using Euler\u2019s method with [latex]n=5[\/latex] steps over the interval [latex]t=\\left[0,1\\right][\/latex]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler\u2019s method. How accurate is Euler\u2019s method?<\/strong><\/p>\n<ol dir=\"auto\" start=\"15\">\n<li>[latex]y^{\\prime} ={t}^{2}[\/latex]<\/li>\n<li>[latex]{y}^{\\prime }=y+{t}^{2},y\\left(0\\right)=3[\/latex]. Exact solution is [latex]y=5{e}^{t}-2-{t}^{2}-2t[\/latex]<\/li>\n<li>[latex]y^{\\prime} ={e}^{\\left(x+y\\right)},y\\left(0\\right)=-1[\/latex]. Exact solution is [latex]y=\\text{-}\\text{ln}\\left(e+1-{e}^{x}\\right)[\/latex]<\/li>\n<li>[latex]{y}^{\\prime }={2}^{x},y\\left(0\\right)=0[\/latex]. Exact solution is [latex]y=\\dfrac{{2}^{x}-1}{\\text{ln}\\left(2\\right)}[\/latex]<\/li>\n<li>[latex]{y}^{\\prime }=-5t,y\\left(0\\right)=-2[\/latex]. Exact solution is [latex]y=-\\dfrac{5}{2}{t}^{2}-2[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following exercises (20-21), consider the initial-value problem [latex]y^{\\prime} =-2y,y\\left(0\\right)=2[\/latex].<\/strong><\/p>\n<ol dir=\"auto\" start=\"20\">\n<li>Draw the directional field of this differential equation.<\/li>\n<li>By calculator or computer, approximate the solution using Euler\u2019s Method at [latex]t=10[\/latex] using [latex]h=100[\/latex].<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Separation of Variables<\/span><\/h2>\n<p dir=\"auto\"><strong>For the following exercises (1-2), solve the following initial-value problems with the initial condition [latex]{y}_{0}=0[\/latex] and graph the solution.<\/strong><\/p>\n<ol dir=\"auto\">\n<li>[latex]\\dfrac{dy}{dt}=y+1[\/latex]<\/li>\n<li>[latex]\\dfrac{dy}{dt}=\\text{-}y - 1[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following exercises (3-7), find the general solution to the differential equation.<\/strong><\/p>\n<ol dir=\"auto\" start=\"3\">\n<li>[latex]{x}^{2}y^{\\prime} =\\left(x+1\\right)y[\/latex]<\/li>\n<li>[latex]y^{\\prime} =2x{y}^{2}[\/latex]<\/li>\n<li>[latex]2x\\dfrac{dy}{dx}={y}^{2}[\/latex]<\/li>\n<li>[latex]\\left(1+x\\right)y^{\\prime} =\\left(x+2\\right)\\left(y - 1\\right)[\/latex]<\/li>\n<li>[latex]t\\dfrac{dy}{dt}=\\sqrt{1-{y}^{2}}[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following exercises (8-12), find the solution to the initial-value problem.<\/strong><\/p>\n<ol dir=\"auto\" start=\"8\">\n<li>[latex]y^{\\prime} ={e}^{y-x},y\\left(0\\right)=0[\/latex]<\/li>\n<li>[latex]\\dfrac{dy}{dx}={y}^{3}x{e}^{{x}^{2}},y\\left(0\\right)=1[\/latex]<\/li>\n<li>[latex]y^{\\prime} =\\dfrac{x}{{\\text{sech}}^{2}y},y\\left(0\\right)=0[\/latex]<\/li>\n<li>[latex]\\dfrac{dx}{dt}=\\text{ln}\\left(t\\right)\\sqrt{1-{x}^{2}},x\\left(1\\right)=0[\/latex]<\/li>\n<li>[latex]y^{\\prime} ={e}^{y}{5}^{x},y\\left(0\\right)=\\text{ln}\\left(\\text{ln}\\left(5\\right)\\right)[\/latex]<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following problems (13-15), use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?<\/strong><\/p>\n<ol dir=\"auto\" start=\"13\">\n<li>[latex]y^{\\prime} =1 - 2y[\/latex]<\/li>\n<li>[latex]y^{\\prime} ={y}^{3}{e}^{x}[\/latex]<\/li>\n<li>[latex]y^{\\prime} =y\\text{ln}\\left(x\\right)[\/latex]<\/li>\n<\/ol>\n<p><strong>For the following exercises (16-17), solve each problem.<\/strong><\/p>\n<ol dir=\"auto\" start=\"16\">\n<li>A drug is administered intravenously to a patient at a rate [latex]r[\/latex] mg\/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, [latex]d[\/latex]. Set up and solve the differential equation, assuming there is no drug initially present in the body.<\/li>\n<li>A tank contains [latex]1[\/latex] kilogram of salt dissolved in [latex]100[\/latex] liters of water. A salt solution of [latex]0.1[\/latex] kg salt\/L is pumped into the tank at a rate of [latex]2[\/latex] L\/min and is drained at the same rate. Solve for the salt concentration at time [latex]t[\/latex]. Assume the tank is well mixed.<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following problems (20-22), use Newton\u2019s law of cooling.<\/strong><\/p>\n<ol dir=\"auto\" start=\"20\">\n<li>The liquid base of an ice cream has an initial temperature of [latex]210^\\circ\\text{F}[\/latex] before it is placed in a freezer with a constant temperature of [latex]20^\\circ\\text{F}\\text{.}[\/latex] After [latex]2[\/latex] hours, the temperature of the ice-cream base has decreased to [latex]170^\\circ\\text{F}\\text{.}[\/latex] At what time will the ice cream be ready to eat? (Assume [latex]30^\\circ\\text{F}[\/latex] is the optimal eating temperature.)<\/li>\n<li>You have a cup of coffee at temperature [latex]70^\\circ\\text{C}[\/latex] and the ambient temperature in the room is [latex]20^\\circ\\text{C}\\text{.}[\/latex] Assuming a cooling rate [latex]k\\text{ of }0.125[\/latex], write and solve the differential equation to describe the temperature of the coffee with respect to time.<\/li>\n<li>You have a cup of coffee at temperature [latex]70^\\circ\\text{C}[\/latex] and you immediately pour in [latex]1[\/latex] part milk to [latex]5[\/latex] parts coffee. The milk is initially at temperature [latex]1^\\circ\\text{C}\\text{.}[\/latex] Write and solve the differential equation that governs the temperature of this coffee.<\/li>\n<\/ol>\n<p dir=\"auto\"><strong>For the following exercises (23-25), solve each problem.<\/strong><\/p>\n<ol dir=\"auto\" start=\"23\">\n<li>Solve the generic problem [latex]y^{\\prime} =ay+b[\/latex] with initial condition [latex]y\\left(0\\right)=c[\/latex].<\/li>\n<li>Assume an initial nutrient amount of [latex]I[\/latex] kilograms in a tank with [latex]L[\/latex] liters. Assume a concentration of [latex]c[\/latex] kg\/L being pumped in at a rate of [latex]r[\/latex] L\/min. The tank is well mixed and is drained at a rate of [latex]r[\/latex] L\/min. Find the equation describing the amount of nutrient in the tank.<\/li>\n<li>Leaves accumulate on the forest floor at a rate of [latex]4[\/latex] g\/cm\u00b2\/yr. These leaves decompose at a rate of [latex]10\\text{%}[\/latex] per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">First-Order Linear Equations and Applications<\/span><\/h2>\n<p dir=\"ltr\" data-pm-slice=\"1 3 []\"><strong>Are the following differential equations linear (1-2)? Explain your reasoning.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" data-tight=\"true\">\n<li>[latex]\\dfrac{dy}{dt}=ty[\/latex]<\/li>\n<li>[latex]y^{\\prime} ={x}^{3}+{e}^{x}[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Write the following first-order differential equations in standard form (3-5).<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"3\" data-tight=\"true\">\n<li>[latex]y^{\\prime} ={x}^{3}y+\\sin{x}[\/latex]<\/li>\n<li>[latex]\\text{-}xy^{\\prime} =\\left(3x+2\\right)y+x{e}^{x}[\/latex]<\/li>\n<li>[latex]\\dfrac{dy}{dt}=yx\\left(x+1\\right)[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>What are the integrating factors for the following differential equations (6-7)?<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"6\" data-tight=\"true\">\n<li>[latex]y^{\\prime} +{e}^{x}y=\\sin{x}[\/latex]<\/li>\n<li>[latex]\\dfrac{dy}{dx}=\\text{tanh}\\left(x\\right)y+1[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Solve the following differential equations (8-12) by using integrating factors.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"8\" data-tight=\"true\">\n<li>[latex]y^{\\prime} =3y+2[\/latex]<\/li>\n<li>[latex]xy^{\\prime} =3y - 6{x}^{2}[\/latex]<\/li>\n<li>[latex]y^{\\prime} =3x+xy[\/latex]<\/li>\n<li>[latex]\\sin\\left(x\\right)y^{\\prime} =y+2x[\/latex]<\/li>\n<li>[latex]xy^{\\prime} =3y+{x}^{2}[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Solve the following differential equations (13-16). Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"13\" data-tight=\"true\">\n<li>[latex]\\left(x+2\\right)y^{\\prime} =2y - 1[\/latex]<\/li>\n<li>[latex]xy^{\\prime} +\\dfrac{y}{2}=\\sin\\left(3t\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+1\\right)y^{\\prime} =3y+{x}^{2}+2x+1[\/latex]<\/li>\n<li>[latex]\\sqrt{{x}^{2}+1}y^{\\prime} =y+2[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Solve the following initial-value problems (17-21) by using integrating factors.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"17\" data-tight=\"true\">\n<li>[latex]y^{\\prime} +y=x,y\\left(0\\right)=3[\/latex]<\/li>\n<li>[latex]xy^{\\prime} =y - 3{x}^{3},y\\left(1\\right)=0[\/latex]<\/li>\n<li>[latex]\\left(1+{x}^{2}\\right)y^{\\prime} =y - 1,y\\left(0\\right)=0[\/latex]<\/li>\n<li>[latex]\\left(2+x\\right)y^{\\prime} =y+2+x,y\\left(0\\right)=0[\/latex]<\/li>\n<li>[latex]\\sqrt{x}y^{\\prime} =y+2x,y\\left(0\\right)=1[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>For the following exercises (22-24), solve each problem.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"22\" data-tight=\"true\">\n<li>A falling object of mass [latex]m[\/latex] can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity given an initial velocity of [latex]0[\/latex].<\/li>\n<li>Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)<\/li>\n<li>Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8[\/latex] m\/s\u00b2 and the proportionality constant is [latex]4?[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>For the following problems (25-26), determine how parameter [latex]a[\/latex] affects the solution.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"25\" data-tight=\"true\">\n<li>Solve the generic equation [latex]y^{\\prime} =ax+y[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/li>\n<li>Solve the generic equation [latex]y^{\\prime} =ax+xy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/li>\n<\/ol>\n<p dir=\"ltr\" data-pm-slice=\"1 3 []\"><strong>For the following problem, consider the logistic equation in the form [latex]P\\prime =CP-{P}^{2}[\/latex]. Find the stability of the equilibria.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"27\" data-tight=\"true\">\n<li>\n<p dir=\"ltr\">[latex]C=0[\/latex]<\/p>\n<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Solve the logistic equation for the given conditions.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"28\" data-tight=\"true\">\n<li>\n<p dir=\"ltr\">Solve the logistic equation for [latex]C=10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/p>\n<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>For the following exercises (29-34), solve each problem.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"29\" data-tight=\"true\">\n<li>A population of deer inside a park has a carrying capacity of [latex]200[\/latex] and a growth rate of [latex]2\\text{%}[\/latex]. If the initial population is [latex]50[\/latex] deer, what is the population of deer at any given time?<\/li>\n<li>Bacteria grow at a rate of [latex]20\\text{%}[\/latex] per hour in a petri dish. If there is initially one bacterium and a carrying capacity of [latex]1[\/latex] million cells, how long does it take to reach [latex]500,000[\/latex] cells?<\/li>\n<li>Two monkeys are placed on an island. After [latex]5[\/latex] years, there are [latex]8[\/latex] monkeys, and the estimated carrying capacity is [latex]25[\/latex] monkeys. When does the population of monkeys reach [latex]16[\/latex] monkeys?<\/li>\n<li>It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant [latex]k[\/latex], as [latex]P\\prime =0.4P\\left(1-\\dfrac{P}{10000}\\right)-kP[\/latex]. Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]2000[\/latex] fish.<\/li>\n<li>Lets add in a minimal threshold value for the species to survive, [latex]T[\/latex], which changes the differential equation to [latex]P\\prime \\left(t\\right)=rP\\left(1-\\dfrac{P}{K}\\right)\\left(1-\\dfrac{T}{P}\\right)[\/latex].\u00a0Bengal tigers in a conservation park have a carrying capacity of [latex]100[\/latex] and need a minimum of [latex]10[\/latex] to survive. If they grow in population at a rate of [latex]1\\text{%}[\/latex] per year, with an initial population of [latex]15[\/latex] tigers, solve for the number of tigers present.<\/li>\n<li>The population of mountain lions in Northern Arizona has an estimated carrying capacity of [latex]250[\/latex] and grows at a rate of [latex]0.25\\text{%}[\/latex] per year and there must be [latex]25[\/latex] for the population to survive. With an initial population of [latex]30[\/latex] mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least [latex]100[\/latex]?)**<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>The following questions (35-37) consider the <span class=\"no-emphasis\" data-type=\"term\">Gompertz equation<\/span>, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.<\/strong><\/p>\n<ol class=\"tight\" dir=\"ltr\" start=\"35\" data-tight=\"true\">\n<li>The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day [latex]1[\/latex] and assuming [latex]\\alpha =0.1[\/latex] and a carrying capacity of [latex]10[\/latex] million cells, how long does it take to reach &#8220;detection&#8221; stage at [latex]5[\/latex] million cells?<\/li>\n<li>It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached [latex]7[\/latex] billion. Was logistic growth or Gompertz growth more accurate, considering world population reached [latex]7[\/latex] billion on October [latex]31,2011?[\/latex]<\/li>\n<li>When does population increase the fastest in the threshold logistic equation [latex]P\\prime \\left(t\\right)=rP\\left(1-\\dfrac{P}{K}\\right)\\left(1-\\dfrac{T}{P}\\right)?[\/latex]<\/li>\n<\/ol>\n<p dir=\"ltr\"><strong>Below is a table of the populations of whooping cranes in the wild from [latex]1940\\text{ to }2000[\/latex]. The population rebounded from near extinction after conservation efforts began. The following problems (38-41) consider applying population models to fit the data. Assume a carrying capacity of [latex]10,000[\/latex] cranes. Fit the data assuming years since [latex]1940[\/latex] (so your initial population at time [latex]0[\/latex] would be [latex]22[\/latex] cranes).<\/strong><\/p>\n<table id=\"fs-id1170571715330\" class=\"unnumbered\" summary=\"A table with eight rows and two columns. The first column has the label\" data-label=\"\">\n<caption><em data-effect=\"italics\">Source:<\/em> https:\/\/www.savingcranes.org\/images\/stories\/site_images\/conservation\/whooping_crane\/pdfs\/historic_wc_numbers.pdf<\/caption>\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Year (years since conservation began)<\/th>\n<th data-align=\"left\">Whooping Crane Population<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1940\\left(0\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]22[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1950\\left(10\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]31[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1960\\left(20\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]36[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1970\\left(30\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]57[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1980\\left(40\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]91[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1990\\left(50\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]159[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2000\\left(60\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]256[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol dir=\"ltr\" start=\"38\">\n<li>Find the equation and parameter [latex]r[\/latex] that best fit the data for the logistic equation.<\/li>\n<li>Find the equation and parameters [latex]r[\/latex] and [latex]T[\/latex] that best fit the data for the threshold logistic equation.<\/li>\n<li>Find the equation and parameter [latex]\\alpha[\/latex] that best fit the data for the Gompertz equation.<\/li>\n<li>Using the three equations found in the previous problems, estimate the population in [latex]2010[\/latex] (year [latex]70[\/latex] after conservation). The real population measured at that time was [latex]437[\/latex]. Which model is most accurate?<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":29,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":669,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/845"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/845\/revisions"}],"predecessor-version":[{"id":1759,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/845\/revisions\/1759"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/845\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=845"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=845"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=845"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}