4.2E: Exercises for Section 8.1

In exercises 1 - 7, determine the order of each differential equation.

1) \(y′+y=3y^2\)

Answer

1st-order

2) \((y′)^2=y′+2y\)

3) \(y'''+y''y′=3x^2\)

Answer

3rd-order

4) \(y′=y''+3t^2\)

5) \(\dfrac{dy}{dt}=t\)

Answer

1st-order

6) \(\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}=3x^4\)

7) \(\left(\dfrac{dy}{dt}\right)^2+8\dfrac{dy}{dt}+3y=4t\)

Answer

1st-order

In exercises 8 - 17, verify that the given function is a solution to the given differential equation.

8) \(y=\dfrac{x^3}{3}\quad\) solves \(\quad y′=x^2\)

9) \(y=2e^{−x}+x−1\quad\) solves \(\quad y′=x−y\)

10) \(y=e^{3x}−\dfrac{e^x}{2}\quad\) solves \(\quad y′=3y+e^x\)

11) \(y=\dfrac{1}{1−x}\quad\) solves \(\quad y′=y^2\)

12) \(y=\dfrac{e^{x^2}}{2}\quad\) solves \(\quad y′=2xy\)

13) \(y=4+\ln x\quad\) solves \(\quad xy′=1\)

14) \(y=3−x+x\ln x\quad\) solves \(\quad y′=\ln x\)

15) \(y=2e^x−x−1\quad\) solves \(\quad y′=y+x\)

16) \(y=e^x+\dfrac{\sin x}{2}−\dfrac{\cos x}{2}\quad\) solves \(\quad y′=\cos x+y\)

17) \(y=πe^{−\cos x}\quad\) solves \(\quad y′=y\sin x\)

In exercises 18 - 27, verify the given general solution and find the particular solution.

18) Find the particular solution to the differential equation \(y′=4x^2\) that passes through \((−3,−30)\), given that \(y=C+\dfrac{4x^3}{3}\) is a general solution.

19) Find the particular solution to the differential equation \(y′=3x^3\) that passes through \((1,4.75)\), given that \(y=C+\dfrac{3x^4}{4}\) is a general solution.

Answer

\(y=4+\dfrac{3x^4}{4}\)

20) Find the particular solution to the differential equation \(y′=3x^2y\) that passes through \((0,12)\), given that \(y=Ce^{x^3}\) is a general solution.

21) Find the particular solution to the differential equation \(y′=2xy\) that passes through \(\left(0,\frac{1}{2}\right)\), given that \(y=Ce^{x^2}\) is a general solution.

Answer

\(y=\frac{1}{2}e^{x^2}\)

22) Find the particular solution to the differential equation \(y′=\big(2xy\big)^2\) that passes through \(\left(1,−\frac{1}{2}\right)\), given that \(y=−\dfrac{3}{C+4x^3}\) is a general solution.

23) Find the particular solution to the differential equation \(y′x^2=y\) that passes through \(\left(1,\frac{2}{e}\right)\), given that \(y=Ce^{−1/x}\) is a general solution.

Answer

\(y=2e^{−1/x}\)

24) Find the particular solution to the differential equation \(8\dfrac{dx}{dt}=−2\cos(2t)−\cos(4t)\) that passes through \((π,π)\), given that \(x=C−\frac{1}{8}\sin(2t)−\frac{1}{32}\sin(4t)\) is a general solution.

25) Find the particular solution to the differential equation \(\dfrac{du}{dt}=\tan u\) that passes through \(\left(1,\frac{π}{2}\right)\), given that \(u=\sin^{−1}\big(e^{C+t}\big)\) is a general solution.

Answer

\(u=\sin^{−1}\big(e^{−1+t}\big)\)

26) Find the particular solution to the differential equation \(\dfrac{dy}{dt}=e^{t+y}\) that passes through \((1,0)\), given that \(y=−\ln(C−e^t)\) is a general solution.

27) Find the particular solution to the differential equation \(y′(1−x^2)=1+y\) that passes through \((0,−2),\) given that \(y=C\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\) is a general solution.

Answer

\(y=−\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\)

In exercises 28 - 37, find the general solution to the differential equation.

28) \(y′=3x+e^x\)

29) \(y′=\ln x+\tan x\)

Answer

\(y=C−x+x\ln x−\ln(\cos x)\)

30) \(y′=\sin x e^{\cos x}\)

31) \(y′=4^x\)

Answer

\(y=C+\dfrac{4^x}{\ln 4}\)

32) \(y′=\sin^{−1}(2x)\)

33) \(y′=2t\sqrt{t^2+16}\)

Answer

\(y=\frac{2}{3}\sqrt{t^2+16}\big(t^2+16\big)+C\)

34) \(x′=\coth t+\ln t+3t^2\)

35) \(x′=t\sqrt{4+t}\)

Answer

\(x=\frac{2}{15}\sqrt{4+t}\big(3t^2+4t−32\big)+C\)

36) \(y′=y\)

37) \(y′=\dfrac{y}{x}\)

Answer

\(y=Cx\)

In exercises 38 - 42, solve the initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=−1\). Draw both solutions on the same graph.

38) \(\dfrac{dy}{dt}=2t\)

39) \(\dfrac{dy}{dt}=−t\)

Answer

\(y=1−\dfrac{t^2}{2},\) and \(y=−\dfrac{t^2}{2}−1\)

40) \(\dfrac{dy}{dt}=2y\)

41) \(\dfrac{dy}{dt}=−y\)

Answer

\(y=e^{−t}\) and \(y=−e^{−t}\)

42) \(\dfrac{dy}{dt}=2\)

In exercises 43 - 47, solve the initial-value problems starting from \(y_0=10\). At what time does \(y\) increase to 100 or drop to 1?

43) \(\dfrac{dy}{dt}=4t\)

Answer

\(y=2(t^2+5),\) When \(t=3\sqrt{5},\) \(y\) will increase to \(100\).

44) \(\dfrac{dy}{dt}=4y\)

45) \(\dfrac{dy}{dt}=−2y\)

Answer

\(y=10e^{−2t},\) When \(t=−\frac{1}{2}\ln\left(\frac{1}{10}\right),\) \(y\) will decrease to \(1\).

46) \(\dfrac{dy}{dt}=e^{4t}\)

47) \(\dfrac{dy}{dt}=e^{−4t}\)

Answer

\(y=\frac{1}{4}(41−e^{−4t}),\) Neither condition will ever happen.

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For exercises 48 - 52, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=−10\) to \(y(t=0)=10\) increasing by 2. Is there some critical point where the behavior of the solution begins to change?

48) [T] \(y′=y(x)\)

49) [T] \(xy′=y\)

Answer

Solution changes from increasing to decreasing at \(y(0)=0\).

50) [T] \(y′=t^3\)

51) [T] \(y′=x+y\) (Hint: \(y=Ce^x−x−1\) is the general solution)

Answer

Solution changes from increasing to decreasing at \(y(0)=0\).

52) [T] \(y′=x\ln x+\sin x\)

53) Find the general solution to describe the velocity of a ball of mass \(1\) lb that is thrown upward at a rate of \(a\) ft/sec.

Answer

\(v(t)=−32t+a\)

54) In the preceding problem, if the initial velocity of the ball thrown into the air is \(a=25\) ft/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.

55) You throw two objects with differing masses \(m_1\) and \(m_2\) upward into the air with the same initial velocity of \(a\) ft/s. What is the difference in their velocity after \(1\) second?

Answer

\(0\) ft/s

56) [T] You throw a ball of mass \(1\) kilogram upward with a velocity of \(a=25\) m/s on Mars, where the force of gravity is \(g=−3.711\) m/s². Use your calculator to approximate how much longer the ball is in the air on Mars.

57) [T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars.

Answer

\(4.86\) meters

58) [T] A car on the freeway accelerates according to \(a=15\cos(πt),\) where \(t\) is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of \(51\) mph. After \(40\) minutes of driving, what is the driver’s velocity?

59) [T] For the car in the preceding problem, find the expression for the distance the car has traveled in time \(t\), assuming an initial distance of \(0\). How long does it take the car to travel \(100\) miles? Round your answer to hours and minutes.

Answer

\(x=50t−\frac{15}{π^2}\cos(πt)+\frac{3}{π^2},2\) hours \(1\) minute

60) [T] For the previous problem, find the total distance traveled in the first hour.

61) Substitute \(y=Be^{3t}\) into \(y′−y=8e^{3t}\) to find a particular solution.

Answer

\(y=4e^{3t}\)

62) Substitute \(y=a\cos(2t)+b\sin(2t)\) into \(y′+y=4\sin(2t)\) to find a particular solution.

63) Substitute \(y=a+bt+ct^2\) into \(y′+y=1+t^2\) to find a particular solution.

Answer

\(y=1−2t+t^2\)

64) Substitute \(y=ae^t\cos t+be^t\sin t\) into \(y′=2e^t\cos t\) to find a particular solution.

65) Solve \(y′=e^{kt}\) with the initial condition \(y(0)=0\) and solve \(y′=1\) with the same initial condition. As \(k\) approaches \(0\), what do you notice?

Answer

\(y=\frac{1}{k}(e^{kt}−1)\) and \(y=t\)