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 $y=t$