Introduction to Differential Equations: Get Stronger Answer Key

Basics of Differential Equations

  1. [latex]1[/latex]
  2. [latex]3[/latex]
  3. [latex]1[/latex]
  4. [latex]1[/latex]
  10. [latex]y=4+\dfrac{3{x}^{4}}{4}[/latex]
  11. [latex]y=\dfrac{1}{2}{e}^{{x}^{2}}[/latex]
  12. [latex]y=2{e}^{\dfrac{\text{-}1}{x}}[/latex]
  13. [latex]u={\sin}^{-1}\left({e}^{-1+t}\right)[/latex]
  14. [latex]y=-\dfrac{\sqrt{x+1}}{\sqrt{1-x}}-1[/latex]
  15. [latex]y=C-x+x\text{ln}x-\text{ln}\left(\cos{x}\right)[/latex]
  16. [latex]y=C+\dfrac{{4}^{x}}{\text{ln}\left(4\right)}[/latex]
  17. [latex]y=\dfrac{2}{3}\sqrt{{t}^{2}+16}\left({t}^{2}+16\right)+C[/latex]
  18. [latex]x=\dfrac{2}{15}\sqrt{4+t}\left(3{t}^{2}+4t - 32\right)+C[/latex]
  19. [latex]y=Cx[/latex]
  20. [latex]y=1-\dfrac{{t}^{2}}{2},y=-\dfrac{{t}^{2}}{2}-1[/latex]
  21. [latex]y={e}^{\text{-}t},y=\text{-}{e}^{\text{-}t}[/latex]
  22. [latex]y=2\left({t}^{2}+5\right),t=3\sqrt{5}[/latex]
  23. [latex]y=10{e}^{-2t},t=-\dfrac{1}{2}\text{ln}\left(\dfrac{1}{10}\right)[/latex]
  24. [latex]y=\dfrac{1}{4}\left(41-{e}^{-4t}\right)[/latex], never
  25. Solution changes from increasing to decreasing at [latex]y\left(0\right)=0[/latex]
  26. Solution changes from increasing to decreasing at [latex]y\left(0\right)=0[/latex]
  27. [latex]v\left(t\right)=-32t+a[/latex]
  28. [latex]0[/latex] ft/s
  29. [latex]y=4{e}^{3t}[/latex]
  30. [latex]y=3 - 2t+{t}^{2}[/latex]
  31. [latex]y=\dfrac{1}{k}\left({e}^{kt}-1\right)[/latex] and [latex]y=x[/latex]

Direction Fields and Euler’s Method

  1. A graph of the given direction field with a flat line drawn on the axis. The arrows point up for y < 0 and down for y > 0. The closer they are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they become.
  2. [latex]y=0[/latex] is a stable equilibrium
  3. A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y > 2 and for y < 0. The arrows point down for 0 < y < 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are. A solution is sketched that follows y = 2 in quadrant two, goes through (0, 1), and then follows the x axis.
  4. [latex]y=0[/latex] is a stable equilibrium and [latex]y=2[/latex] is unstable
  5. A direction field over the four quadrants. As t goes from 0 to infinity, the arrows become more and more vertical after being horizontal closer to x = 0.
  6. A direction field over [-2, 2] in the x and y axes. The arrows point slightly down and to the right over [-2, 0] and gradually become vertical over [0, 2].
  7. A direction field with horizontal arrows pointing to the right at y = 1 and y = -1. The arrows point up for y < -1 and y > 1. The arrows point down for -1 < y < 1. The closer the arrows are to these lines, the more horizontal they are, and the further away they are, the more vertical they are.
  8. A direction field with arrows pointing down and to the right for nearly all points in [-2, 2] on the x and y axes. Close to the origin, the arrows become more horizontal, point to the upper right, become more horizontal, and then point down to the right again.
  9. A direction field with horizontal arrows pointing to the right on the x-axis and x = -3. Above the x-axis and for x < -3, the arrows point down. For x > -3, the arrows point up. Below the x-axis and for x < -3, the arrows point up. For x > -3, the arrows point down. The further away from the x-axis and x = -3, the arrows become more vertical, and the closer they become, the more horizontal they become.
  10. E
  11. A
  12. B
  13. A
  14. C
  15. [latex]2.24[/latex], exact: [latex]3[/latex]
  16. [latex]7.739364[/latex], exact: [latex]5\left(e - 1\right)[/latex]
  17. [latex]-0.2535[/latex], exact: [latex]0[/latex]
  18. [latex]1.345[/latex], exact: [latex]\dfrac{1}{\text{ln}\left(2\right)}[/latex]
  19. [latex]-4[/latex], exact: [latex]\dfrac{\text{-}1}{2}[/latex]
  20. A direction field with horizontal arrows pointing to the right on the x axis. Above the x axis, the arrows point down and to the right. Below the x axis, the arrows point up and to the right. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are from the x axis, the more vertical the arrows are.
  21. [latex]4.0741{e}^{-10}[/latex]

Separation of Variables

  1. [latex]y={e}^{t}-1[/latex]
  2. [latex]y=1-{e}^{\text{-}t}[/latex]
  3. [latex]y=Cx{e}^{\dfrac{-1}{x}}[/latex]
  4. [latex]y=\dfrac{1}{C-{x}^{2}}[/latex]
  5. [latex]y=-\dfrac{2}{C+\text{ln}x}[/latex]
  6. [latex]y=C{e}^{x}\left(x+1\right)+1[/latex]
  7. [latex]y=\sin\left(\text{ln}t+C\right)[/latex]
  8. [latex]y=\text{-}\text{ln}\left({e}^{\text{-}x}\right)[/latex]
  9. [latex]y=\dfrac{1}{\sqrt{2-{e}^{{x}^{2}}}}[/latex]
  10. [latex]y={\text{tanh}}^{-1}\left(\dfrac{{x}^{2}}{2}\right)[/latex]
  11. [latex]x=\sin\left(1-t+t\text{ln}t\right)[/latex]
  12. [latex]y=\text{ln}\left(\text{ln}\left(5\right)\right)-\text{ln}\left(2-{5}^{x}\right)[/latex]
  13. [latex]y=C{e}^{-2x}+\dfrac{1}{2}[/latex]
  14. [latex]y=\dfrac{1}{\sqrt{2}\sqrt{C-{e}^{x}}}[/latex]
  15. [latex]y=C{e}^{\text{-}x}{x}^{x}[/latex]
  16. [latex]y=\dfrac{r}{d}\left(1-{e}^{\text{-}dt}\right)[/latex]
  17. [latex]y\left(t\right)=10 - 9{e}^{\dfrac{\text{-}x}{50}}[/latex]
  18. [latex]T\left(t\right)=20+50{e}^{-0.125t}[/latex]
  19. [latex]T\left(t\right)=20+38.5{e}^{-0.125t}[/latex]
  20. [latex]24[/latex] hours [latex]57[/latex] minutes
  21. [latex]T\left(t\right)=20+50{e}^{-0.125t}[/latex]
  22. [latex]T\left(t\right)=20+38.5{e}^{-0.125t}[/latex]
  23. [latex]y=\left(c+\dfrac{b}{a}\right){e}^{ax}-\dfrac{b}{a}[/latex]
  24. [latex]y\left(t\right)=cL+\left(I-cL\right){e}^{\dfrac{\text{-}rt}{L}}[/latex]
  25. [latex]y=40\left(1-{e}^{-0.1t}\right),40[/latex] g/cm²

First-Order Linear Equations and Applications

  1. Yes
  2. Yes
  3. [latex]y^{\prime} -{x}^{3}y=\sin{x}[/latex]
  4. [latex]y^{\prime} +\dfrac{\left(3x+2\right)}{x}y=\text{-}{e}^{x}[/latex]
  5. [latex]\dfrac{dy}{dt}-yx\left(x+1\right)=0[/latex]
  6. [latex]{e}^{x}[/latex]
  7. [latex]\text{-}\text{ln}\left(\text{cosh}x\right)[/latex]
  8. [latex]y=C{e}^{3x}-\dfrac{2}{3}[/latex]
  9. [latex]y=C{x}^{3}+6{x}^{2}[/latex]
  10. [latex]y=C{e}^{\dfrac{{x}^{2}}{2}}-3[/latex]
  11. [latex]y=C\tan\left(\dfrac{x}{2}\right)-2x+4\tan\left(\dfrac{x}{2}\right)\text{ln}\left(\sin\left(\dfrac{x}{2}\right)\right)[/latex]
  12. [latex]y=C{x}^{3}-{x}^{2}[/latex]
  13. [latex]y=C{\left(x+2\right)}^{2}+\dfrac{1}{2}[/latex]
  14. [latex]y=\dfrac{C}{\sqrt{x}}+2\sin\left(3t\right)[/latex]
  15. [latex]y=C{\left(x+1\right)}^{3}-{x}^{2}-2x - 1[/latex]
  16. [latex]y=C{e}^{{\text{sinh}}^{-1}x}-2[/latex]
  17. [latex]y=x+4{e}^{x}-1[/latex]
  18. [latex]y=-\dfrac{3x}{2}\left({x}^{2}-1\right)[/latex]
  19. [latex]y=1-{e}^{{\tan}^{-1}x}[/latex]
  20. [latex]y=\left(x+2\right)\text{ln}\left(\dfrac{x+2}{2}\right)[/latex]
  21. [latex]y=2{e}^{2\sqrt{x}}-2x - 2\sqrt{x}-1[/latex]
  22. [latex]v\left(t\right)=\dfrac{gm}{k}\left(1-{e}^{\dfrac{\text{-}kt}{m}}\right)[/latex]
  23. [latex]\sqrt{\dfrac{gm}{k}}[/latex]
  24. [latex]40.451[/latex] seconds
  25. [latex]y=C{e}^{x}-a\left(x+1\right)[/latex]
  26. [latex]y=C{e}^{\dfrac{{x}^{2}}{2}}-a[/latex]
  27. [latex]P=0[/latex] semi-stable
  28. [latex]P=\dfrac{10{e}^{10x}}{{e}^{10x}+4}[/latex]
  29. [latex]P\left(t\right)=\dfrac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}[/latex]
  30. [latex]69[/latex] hours [latex]5[/latex] minutes
  31. [latex]8[/latex] years [latex]11[/latex] months
  32. [latex]y=\dfrac{-20}{4\times {10}^{-6}-0.002{e}^{0.01t}}[/latex]
  33. [latex]P\left(t\right)=\dfrac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}[/latex]
  34. [latex]13[/latex] years
  35. [latex]31.465[/latex] days
  36. September [latex]2008[/latex]
  37. [latex]\dfrac{K+T}{2}[/latex]
  38. [latex]r=0.0405[/latex]
  40. [latex]\alpha =0.0081[/latex]
  41. Logistic: [latex]361[/latex], Threshold: [latex]436[/latex], Gompertz: [latex]309[/latex].