
17.1E: Exercises for Section 17.1


In exercises 1 - 6, classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.

1.  $$x^3y''+(x-1)y'-8y=0$$

linear, homogenous

2.  $$(1+y^2)y''+xy'-3y= \cos x$$

3.  $$xy''+e^yy'=x$$

nonlinear

4.  $$y''+ \dfrac{4}{x}y'-8xy=5x^2+1$$

5.  $$y''+( \sin x)y'-xy=4y$$

linear, homogenous

6.  $$y''+\left(\dfrac{x+3}{y}\right)y'=0$$

In exercises 7 - 10, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of $$c_1$$ and $$c_2.$$ What do the solutions have in common?

7. [T]  $$y''+2y'-3y=0; \quad y(x)=c_1e^x+c_2e^{-3x}$$

8. [T]  $$x^2y''-2y-3x^2+1=0; \quad y(x)=c_1x^2+c_2x^{-1}+x^2 \ln(x)+ \frac{1}{2}$$

9. [T]  $$y''+14y+49y=0; \quad y(x)=c_1e^{−7x}+c_2xe^{−7x}$$

10. [T]  $$6y''−49y′+8y=0; \quad y(x)=c_1e^{x/6}+c_2e^{8x}$$

In exercises 11 - 30, find the general solution to the linear differential equation.

11.  $$y''−3y′−10y=0$$

$$y = c_1e^{5x} + c_2e^{-2x}$$

12.  $$y''−7y′+12y=0$$

13.  $$y''+4y′+4y=0$$

$$y = c_1e^{-2x} + c_2xe^{-2x}$$

14.  $$4y''−12y′+9y=0$$

15.  $$2y''−3y′−5y=0$$

$$y = c_1e^{5x/2} + c_2e^{-x}$$

16.  $$3y''−14y′+8y=0$$

17.  $$y''+y′+y=0$$

$$y = e^{-x/2}\left(c_1\cos\frac{\sqrt{3}x}{2} + c_2\sin\frac{\sqrt{3}x}{2}\right)$$

18.  $$5y''+2y′+4y=0$$

19.  $$y''−121y=0$$

$$y = c_1e^{-11x} + c_2e^{11x}$$

20.  $$8y''+14y′−15y=0$$

21.  $$y''+81y=0$$

$$y = c_1\cos 9x + c_2\sin 9x$$

22.  $$y''−y′+11y=0$$

23.  $$2y''=0$$

$$y = c_1 + c_2x$$

24.  $$y''−6y′+9y=0$$

25.  $$3y''−2y′−7y=0$$

$$y = c_1e^{\left( (1+\sqrt{22})/3 \right)x} + c_2e^{\left( (1-\sqrt{22})/3 \right)x}$$

26.  $$4y''−10y′=0$$

27.  $$36\dfrac{d^2y}{dx^2}+12\dfrac{dy}{dx}+y=0$$

$$y = c_1e^{-x/6} + c_2xe^{-x/6}$$

28.  $$25\dfrac{d^2y}{dx^2}−80\dfrac{dy}{dx}+64y=0$$

29.  $$\dfrac{d^2y}{dx^2}−9\dfrac{dy}{dx}=0$$

$$y = c_1 + c_2e^{9x}$$

30.  $$4\dfrac{d^2y}{dx^2}+8y=0$$

In exercises 31 - 38, solve the initial-value problem.

31.  $$y''+5y′+6y=0, \quad y(0)=0,\; y′(0)=−2$$

$$y = -2e^{-2x} + 2e^{-3x}$$

32.  $$y''+2y′−8y=0, \quad y(0)=5,\; y′(0)=4$$

33.  $$y''+4y=0, \quad y(0)=3, \; y′(0)=10$$

$$y = 3\cos(2x) + 5\sin(2x)$$

34.  $$y''−18y′+81y=0, \quad y(0)=1, \; y′(0)=5$$

35.  $$y''−y′−30y=0, \quad y(0)=1, \; y′(0)=−16$$

$$y = -e^{6x} + 2e^{-5x}$$

36.  $$4y''+4y′−8y=0, \quad y(0)=2, \; y′(0)=1$$

37.  $$25y''+10y′+y=0, \quad y(0)=2, \; y′(0)=1$$

$$y = 2e^{-x/5} + \frac{7}{5}xe^{-x/5}$$

38.  $$y''+y=0, \quad y(π)=1, \; y′(π)=−5$$

In exercises 39 - 46, solve the boundary-value problem, if possible.

39.  $$y''+y′−42y=0, \quad y(0)=0, \; y(1)=2$$

$$y = \left( \frac{2}{e^6 - e^{-7}} \right)e^{6x} - \left( \frac{2}{e^6 - e^{-7}} \right)e^{-7x}$$

40.  $$9y''+y=0, \quad y(3π^2)=6, \; y(0)=−8$$

41.  $$y''+10y′+34y=0, \quad y(0)=6, \; y(π)=2$$

No solutions exist.

42.  $$y''+7y′−60y=0, \quad y(0)=4, \; y(2)=0$$

43.  $$y''−4y′+4y=0, \quad y(0)=2, \; y(1)=−1$$

$$y = 2e^{2x} - \left( \frac{2e^{2}+1}{e^2} \right)xe^{2x}$$

44.  $$y''−5y′=0, \quad y(0)=3, \; y(−1)=2$$

45.  $$y''+9y=0, \quad y(0)=4, \; y(π^3)=−4$$

$$y = 4\cos 3x + c_2\sin 3x,$$ infinitely many solutions

46.  $$4y''+25y=0, \quad y(0)=2, \; y(2π)=−2$$

47. Find a differential equation with a general solution that is $$y=c_1e^{x/5}+c_2e^{−4x}.$$

$$5y'' +19y' -4y = 0$$

48. Find a differential equation with a general solution that is $$y=c_1e^{x}+c_2e^{−4x/3}.$$

For each differential equation in exercises 49 - 51:

1. Solve the initial value problem.
2. [T] Use a graphing utility to graph the particular solution.

49.  $$y''+64y=0; \quad y(0)=3, \; y′(0)=16$$

a.  $$y = 3\cos 8x + 2\sin 8x$$
b.

50.  $$y''−2y′+10y=0; \quad y(0)=1, \; y′(0)=13$$

51.  $$y''+5y′+15y=0; \quad y(0)=−2, \; y′(0)=7$$

a.  $$y = e^{-5/2}\left[-2\cos\left(\frac{\sqrt{35}}{2}x\right) + \frac{4\sqrt{35}}{35}\sin\left(\frac{\sqrt{35}}{2}x\right) \right]$$
b.

52. (Principle of superposition) Prove that if $$y_1(x)$$ and $$y_2(x)$$ are solutions to a linear homogeneous differential equation, $$y''+p(x)y′+q(x)y=0,$$ then the function $$y(x)=c_1y_1(x)+c_2y_2(x),$$ where $$c_1$$ and $$c_2$$ are constants, is also a solution.

53. Prove that if $$a, \, b$$ and $$c$$ are positive constants, then all solutions to the second-order linear differential equation $$ay''+by′+cy=0$$ approach zero as $$x→∞.$$ (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

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