# 8.1E: Exercises for Basics of Differential Equations

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

1) $$y′+y=3y^2$$

1st-order

2) $$(y′)^2=y′+2y$$

3) $$y'''+y''y′=3x^2$$

3rd-order

4) $$y′=y''+3t^2$$

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

1st-order

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

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

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 $$\quad y′=xy$$

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.

$$y=4+\dfrac{3x^4}{4}$$

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.

$$y=\frac{1}{2}e^{x^2}$$

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.

$$y=2e^{−1/x}$$

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.

$$u=\sin^{−1}\big(e^{−1+t}\big)$$

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.

$$y=−\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1$$

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

28) $$y′=3x+e^x$$

29) $$y′=\ln x+\tan x$$

$$y=C−x+x\ln x−\ln(\cos x)$$

30) $$y′=\sin x e^{\cos x}$$

31) $$y′=4^x$$

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

32) $$y′=\sin^{−1}(2x)$$

33) $$y′=2t\sqrt{t^2+16}$$

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

34) $$x′=\coth t+\ln t+3t^2$$

35) $$x′=t\sqrt{4+t}$$

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

36) $$y′=y$$

37) $$y′=\dfrac{y}{x}$$

$$y=Cx$$

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) $$\dfrac{dy}{dt}=−t$$

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

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

41) $$\dfrac{dy}{dt}=−y$$

$$y=e^{−t}$$ 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) $$\dfrac{dy}{dt}=4t$$

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

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

45) $$\dfrac{dy}{dt}=−2y$$

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

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

47) $$\dfrac{dy}{dt}=e^{−4t}$$

$$y=\frac{1}{4}(41−e^{−4t}),$$ 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] $$xy′=y$$

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)

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.

$$v(t)=−32t+a$$

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?

$$0$$ 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.

$$4.86$$ 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.

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

$$y=4e^{3t}$$

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.

$$y=1−2t+t^2$$

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?

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