{"id":789,"date":"2025-06-20T17:12:58","date_gmt":"2025-06-20T17:12:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=789"},"modified":"2025-07-29T16:45:29","modified_gmt":"2025-07-29T16:45:29","slug":"introduction-to-differential-equations-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-differential-equations-cheat-sheet\/","title":{"raw":"Introduction to Differential Equations: Cheat Sheet","rendered":"Introduction to Differential Equations: Cheat Sheet"},"content":{"raw":"<strong>Basics of Differential Equations<\/strong>\r\n<ul id=\"fs-id1170571508562\" data-bullet-style=\"bullet\">\r\n \t<li>A differential equation is an equation involving a function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/li>\r\n \t<li>The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.<\/li>\r\n \t<li>A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.<\/li>\r\n<\/ul>\r\n<strong>Direction Fields and Euler's Method<\/strong>\r\n<ul id=\"fs-id1170571042951\" data-bullet-style=\"bullet\">\r\n \t<li>A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.<\/li>\r\n \t<li>Euler\u2019s Method is a numerical technique that can be used to approximate solutions to a differential equation.<\/li>\r\n<\/ul>\r\n<strong>Separation of Variables<\/strong>\r\n<ul id=\"fs-id1170573604134\" data-bullet-style=\"bullet\">\r\n \t<li>A separable differential equation is any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex].<\/li>\r\n \t<li>The method of separation of variables is used to find the general solution to a separable differential equation.<\/li>\r\n<\/ul>\r\n<strong>First-Order Linear Equations and Applications<\/strong>\r\n<ul id=\"fs-id1170572560614\" data-bullet-style=\"bullet\">\r\n \t<li>When studying population functions, different assumptions\u2014such as exponential growth, logistic growth, or threshold population\u2014lead to different rates of growth.<\/li>\r\n \t<li>The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.<\/li>\r\n \t<li>The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.<\/li>\r\n \t<li>Any first-order linear differential equation can be written in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex].<\/li>\r\n \t<li>We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.<\/li>\r\n \t<li>Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170571042973\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Euler\u2019s Method<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\begin{array}{c}{x}_{n}={x}_{0}+nh\\hfill \\\\ {y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right),\\text{where}h\\text{is the step size}\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Separable differential equation<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{y}^{\\prime }=f\\left(x\\right)g\\left(y\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Solution concentration<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{du}{dt}=\\text{INFLOW RATE}-\\text{OUTFLOW RATE}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Newton\u2019s law of cooling<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dT}{dt}=k\\left(T-{T}_{s}\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Logistic differential equation and initial-value problem<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dP}{dt}=rP\\left(1-\\frac{P}{K}\\right),P\\left(0\\right)={P}_{0}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Solution to the logistic differential equation\/initial-value problem<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]P\\left(t\\right)=\\frac{{P}_{0}K{e}^{rt}}{\\left(K-{P}_{0}\\right)+{P}_{0}{e}^{rt}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Threshold population model<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dP}{dt}=\\text{-}rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{P}{T}\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">standard form<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">integrating factor<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\mu \\left(x\\right)={e}^{\\displaystyle\\int p\\left(x\\right)dx}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl>\r\n \t<dt>\r\n<dl id=\"fs-id1170571116853\">\r\n \t<dt>asymptotically semi-stable solution<\/dt>\r\n \t<dd id=\"fs-id1170571116858\">[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571116871\">\r\n \t<dt>asymptotically stable solution<\/dt>\r\n \t<dd id=\"fs-id1170571116885\">[latex]y=k[\/latex] if there exists [latex]\\epsilon &gt;0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571459118\">\r\n \t<dt>asymptotically unstable solution<\/dt>\r\n \t<dd id=\"fs-id1170571459132\">[latex]y=k[\/latex] if there exists [latex]\\epsilon &gt;0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] never approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1170571243486\">\r\n \t<dt>autonomous differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571243492\">an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1170572147813\">\r\n \t<dt>carrying capacity<\/dt>\r\n \t<dd id=\"fs-id1170572351566\">the maximum population of an organism that the environment can sustain indefinitely<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571278839\">\r\n \t<dt>differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571278844\">an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146732\">\r\n \t<dt>\r\n<dl id=\"fs-id1170571051486\">\r\n \t<dt>direction field (slope field)<\/dt>\r\n \t<dd id=\"fs-id1170571051491\">a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571051498\">\r\n \t<dt>equilibrium solution<\/dt>\r\n \t<dd id=\"fs-id1170571042367\">any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042387\">\r\n \t<dt>Euler\u2019s Method<\/dt>\r\n \t<dd id=\"fs-id1170571042393\">a numerical technique used to approximate solutions to an initial-value problem<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>general solution (or family of solutions)<\/dt>\r\n \t<dd id=\"fs-id1170571146738\">the entire set of solutions to a given differential equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572351570\">\r\n \t<dt>growth rate<\/dt>\r\n \t<dd id=\"fs-id1170572351576\">the constant [latex]r&gt;0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572290475\">\r\n \t<dt>initial population<\/dt>\r\n \t<dd id=\"fs-id1170572290480\">the population at time [latex]t=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146742\">\r\n \t<dt>initial value(s)<\/dt>\r\n \t<dd id=\"fs-id1170571146747\">a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146753\">\r\n \t<dt>initial velocity<\/dt>\r\n \t<dd id=\"fs-id1170571146758\">the velocity at time [latex]t=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146771\">\r\n \t<dt>initial-value problem<\/dt>\r\n \t<dd id=\"fs-id1170571146776\">a differential equation together with an initial value or values<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571813852\">\r\n \t<dt>integrating factor<\/dt>\r\n \t<dd id=\"fs-id1170571813858\">any function [latex]f\\left(x\\right)[\/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571813876\">\r\n \t<dt>linear<\/dt>\r\n \t<dd id=\"fs-id1170571813882\">description of a first-order differential equation that can be written in the form [latex]a\\left(x\\right){y}^{\\prime }+b\\left(x\\right)y=c\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571688565\">\r\n \t<dt>logistic differential equation<\/dt>\r\n \t<dd id=\"fs-id1170572624812\">a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146780\">\r\n \t<dt>order of a differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571146786\">the highest order of any derivative of the unknown function that appears in the equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146790\">\r\n \t<dt>particular solution<\/dt>\r\n \t<dd id=\"fs-id1170571146795\">member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379509\">\r\n \t<dt>phase line<\/dt>\r\n \t<dd id=\"fs-id1170572379514\">a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571243500\">\r\n \t<dt>separable differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571243506\">any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573742138\">\r\n \t<dt>separation of variables<\/dt>\r\n \t<dd id=\"fs-id1170573742143\">a method used to solve a separable differential equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042397\">\r\n \t<dt>solution curve<\/dt>\r\n \t<dd id=\"fs-id1170571042402\">a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146801\">\r\n \t<dt>solution to a differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571146806\">a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572386140\">\r\n \t<dt>standard form<\/dt>\r\n \t<dd id=\"fs-id1170572386145\">the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042408\">\r\n \t<dt>step size<\/dt>\r\n \t<dd id=\"fs-id1170571042413\">the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379519\">\r\n \t<dt>threshold population<\/dt>\r\n \t<dd id=\"fs-id1170571638272\">the minimum population that is necessary for a species to survive<\/dd>\r\n<\/dl>","rendered":"<p><strong>Basics of Differential Equations<\/strong><\/p>\n<ul id=\"fs-id1170571508562\" data-bullet-style=\"bullet\">\n<li>A differential equation is an equation involving a function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/li>\n<li>The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.<\/li>\n<li>A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.<\/li>\n<\/ul>\n<p><strong>Direction Fields and Euler&#8217;s Method<\/strong><\/p>\n<ul id=\"fs-id1170571042951\" data-bullet-style=\"bullet\">\n<li>A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.<\/li>\n<li>Euler\u2019s Method is a numerical technique that can be used to approximate solutions to a differential equation.<\/li>\n<\/ul>\n<p><strong>Separation of Variables<\/strong><\/p>\n<ul id=\"fs-id1170573604134\" data-bullet-style=\"bullet\">\n<li>A separable differential equation is any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex].<\/li>\n<li>The method of separation of variables is used to find the general solution to a separable differential equation.<\/li>\n<\/ul>\n<p><strong>First-Order Linear Equations and Applications<\/strong><\/p>\n<ul id=\"fs-id1170572560614\" data-bullet-style=\"bullet\">\n<li>When studying population functions, different assumptions\u2014such as exponential growth, logistic growth, or threshold population\u2014lead to different rates of growth.<\/li>\n<li>The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.<\/li>\n<li>The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.<\/li>\n<li>Any first-order linear differential equation can be written in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex].<\/li>\n<li>We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.<\/li>\n<li>Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170571042973\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Euler\u2019s Method<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\begin{array}{c}{x}_{n}={x}_{0}+nh\\hfill \\\\ {y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right),\\text{where}h\\text{is the step size}\\hfill \\end{array}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Separable differential equation<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{y}^{\\prime }=f\\left(x\\right)g\\left(y\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Solution concentration<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{du}{dt}=\\text{INFLOW RATE}-\\text{OUTFLOW RATE}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Newton\u2019s law of cooling<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dT}{dt}=k\\left(T-{T}_{s}\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Logistic differential equation and initial-value problem<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dP}{dt}=rP\\left(1-\\frac{P}{K}\\right),P\\left(0\\right)={P}_{0}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Solution to the logistic differential equation\/initial-value problem<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]P\\left(t\\right)=\\frac{{P}_{0}K{e}^{rt}}{\\left(K-{P}_{0}\\right)+{P}_{0}{e}^{rt}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Threshold population model<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dP}{dt}=\\text{-}rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{P}{T}\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">standard form<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">integrating factor<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\mu \\left(x\\right)={e}^{\\displaystyle\\int p\\left(x\\right)dx}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl>\n<dt>\n<\/dt>\n<dt>asymptotically semi-stable solution<\/dt>\n<dd id=\"fs-id1170571116858\">[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571116871\">\n<dt>asymptotically stable solution<\/dt>\n<dd id=\"fs-id1170571116885\">[latex]y=k[\/latex] if there exists [latex]\\epsilon >0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571459118\">\n<dt>asymptotically unstable solution<\/dt>\n<dd id=\"fs-id1170571459132\">[latex]y=k[\/latex] if there exists [latex]\\epsilon >0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] never approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571243486\">\n<dt>autonomous differential equation<\/dt>\n<dd id=\"fs-id1170571243492\">an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572147813\">\n<dt>carrying capacity<\/dt>\n<dd id=\"fs-id1170572351566\">the maximum population of an organism that the environment can sustain indefinitely<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571278839\">\n<dt>differential equation<\/dt>\n<dd id=\"fs-id1170571278844\">an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146732\">\n<dt>\n<\/dt>\n<dt>direction field (slope field)<\/dt>\n<dd id=\"fs-id1170571051491\">a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571051498\">\n<dt>equilibrium solution<\/dt>\n<dd id=\"fs-id1170571042367\">any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042387\">\n<dt>Euler\u2019s Method<\/dt>\n<dd id=\"fs-id1170571042393\">a numerical technique used to approximate solutions to an initial-value problem<\/dd>\n<\/dl>\n<p> \tgeneral solution (or family of solutions)<br \/>\n \tthe entire set of solutions to a given differential equation<\/p>\n<dl id=\"fs-id1170572351570\">\n<dt>growth rate<\/dt>\n<dd id=\"fs-id1170572351576\">the constant [latex]r>0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572290475\">\n<dt>initial population<\/dt>\n<dd id=\"fs-id1170572290480\">the population at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146742\">\n<dt>initial value(s)<\/dt>\n<dd id=\"fs-id1170571146747\">a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146753\">\n<dt>initial velocity<\/dt>\n<dd id=\"fs-id1170571146758\">the velocity at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146771\">\n<dt>initial-value problem<\/dt>\n<dd id=\"fs-id1170571146776\">a differential equation together with an initial value or values<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571813852\">\n<dt>integrating factor<\/dt>\n<dd id=\"fs-id1170571813858\">any function [latex]f\\left(x\\right)[\/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571813876\">\n<dt>linear<\/dt>\n<dd id=\"fs-id1170571813882\">description of a first-order differential equation that can be written in the form [latex]a\\left(x\\right){y}^{\\prime }+b\\left(x\\right)y=c\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571688565\">\n<dt>logistic differential equation<\/dt>\n<dd id=\"fs-id1170572624812\">a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146780\">\n<dt>order of a differential equation<\/dt>\n<dd id=\"fs-id1170571146786\">the highest order of any derivative of the unknown function that appears in the equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146790\">\n<dt>particular solution<\/dt>\n<dd id=\"fs-id1170571146795\">member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379509\">\n<dt>phase line<\/dt>\n<dd id=\"fs-id1170572379514\">a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571243500\">\n<dt>separable differential equation<\/dt>\n<dd id=\"fs-id1170571243506\">any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573742138\">\n<dt>separation of variables<\/dt>\n<dd id=\"fs-id1170573742143\">a method used to solve a separable differential equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042397\">\n<dt>solution curve<\/dt>\n<dd id=\"fs-id1170571042402\">a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146801\">\n<dt>solution to a differential equation<\/dt>\n<dd id=\"fs-id1170571146806\">a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572386140\">\n<dt>standard form<\/dt>\n<dd id=\"fs-id1170572386145\">the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042408\">\n<dt>step size<\/dt>\n<dd id=\"fs-id1170571042413\">the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379519\">\n<dt>threshold population<\/dt>\n<dd id=\"fs-id1170571638272\">the minimum population that is necessary for a species to survive<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"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\/789"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/789\/revisions"}],"predecessor-version":[{"id":1572,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/789\/revisions\/1572"}],"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\/789\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=789"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=789"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=789"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}