# 5.3.1: Nonhomgeneous Linear Equations (Exercises)

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## Q5.3.1

In Exercises 5.3.1-5.3.6 find a particular solution by the method used in Example 5.3.2. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.

1. $$y''+5y'-6y=22+18x-18x^2$$

2. $$y''-4y'+5y=1+5x$$

3. $$y''+8y'+7y=-8-x+24x^2+7x^3$$

4. $$y''-4y'+4y=2+8x-4x^2$$

5. $$y''+2y'+10y=4+26x+6x^2+10x^3, \quad y(0)=2, \quad y'(0)=9$$

6. $$y''+6y'+10y=22+20x, \quad y(0)=2,\; y'(0)=-2$$

## Q5.3.2

7. Show that the method used in Example 5.3.2 will not yield a particular solution of

$y''+y'=1+2x+x^2; \tag{A}$

that is, (A) does’nt have a particular solution of the form $$y_p=A+Bx+Cx^2$$, where $$A$$, $$B$$, and $$C$$ are constants.

## Q5.3.3

In Exercises 5.3.8-5.3.13 find a particular solution by the method used in Example 5.3.3.

8. $$x^{2}y'' +7xy'+8y=\frac{6}{x}$$

9. $$x^{2}y''-7xy'+7y=13x^{1/2}$$

10. $$x^{2}y''-xy'+y=2x^{3}$$

11. $$x^{2}y''+5xy'+4y=\frac{1}{x^{3}}$$

12. $$x^{2}y''+xy'+y=10x^{1/3}$$

13. $$x^{2}y''-3xy'+13y=2x^{4}$$

## Q5.3.4

14. Show that the method suggested for finding a particular solution in Exercises 5.3.8-5.3.13 will not yield a particular solution of

$x^2y''+3xy'-3y={1\over x^3}; \tag{A}$

that is, (A) doesn’t have a particular solution of the form $$y_p=A/x^3$$.

15. Prove: If $$a$$, $$b$$, $$c$$, $$\alpha$$, and $$M$$ are constants and $$M\ne0$$ then

$ax^2y''+bxy'+cy=M x^\alpha$

has a particular solution $$y_p=Ax^\alpha$$ ($$A=$$ constant) if and only if $$a\alpha(\alpha-1)+b\alpha+c\ne0$$.

## Q5.3.5

If $$a, b, c,$$ and $$\alpha$$ are constants, then $\alpha (e^{\alpha x})'' +b(e^{\alpha x})'+ce^{\alpha x} = (a\alpha ^{2}+b\alpha + c)e^{\alpha x}.$ Use this in Exercises 5.3.16-5.3.21 to find a particular solution. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.

16. $$y''+5y'-6y=6e^{3x}$$

17. $$y''-4y'+5y=e^{2x}$$

18. $$y''+8y'+7y=10e^{-2x}, \quad y(0)=-2,\; y'(0)=10$$

19. $$y''-4y'+4y=e^{x}, \quad y(0)=2,\quad y'(0)=0$$

20. $$y''+2y'+10y=e^{x/2}$$

21. $$y''+6y'+10y=e^{-3x}$$

## Q5.3.6

22. Show that the method suggested for finding a particular solution in Exercises 5.3.16-5.3.21 will not yield a particular solution of

$y''-7y'+12y=5e^{4x}; \tag{A}$

that is, (A) doesn’t have a particular solution of the form $$y_p=Ae^{4x}$$.

23. Prove: If $$\alpha$$ and $$M$$ are constants and $$M\ne0$$ then constant coefficient equation

$ay''+by'+cy=M e^{\alpha x}$

has a particular solution $$y_p=Ae^{\alpha x}$$ ($$A=$$ constant) if and only if $$e^{\alpha x}$$ isn’t a solution of the complementary equation.

## Q5.3.7

If $$ω$$ is a constant, differentiating a linear combination of $$\cos ωx$$ and $$\sin ωx$$ with respect to $$x$$ yields another linear combination of $$\cos ωx$$ and $$\sin ωx$$. In Exercises 5.3.24-5.3.29 use this to find a particular solution of the equation. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.

24. $$y''-8y'+16y=23\cos x-7\sin x$$

25. $$y''+y'=-8\cos2x+6\sin2x$$

26. $$y''-2y'+3y=-6\cos3x+6\sin3x$$

27. $$y''+6y'+13y=18\cos x+6\sin x$$

28. $$y''+7y'+12y=-2\cos2x+36\sin2x, \quad y(0)=-3,\quad y'(0)=3$$

29. $$y''-6y'+9y=18\cos3x+18\sin3x, \quad y(0)=2,\quad y'(0)=2$$

## Q5.3.8

30. Find the general solution of

$y''+\omega_0^2y =M\cos\omega x+N\sin\omega x,$

where $$M$$ and $$N$$ are constants and $$\omega$$ and $$\omega_0$$ are distinct positive numbers.

31. Show that the method suggested for finding a particular solution in Exercises 5.3.24-5.3.29 will not yield a particular solution of

$y''+y=\cos x+\sin x; \tag{A}$

that is, (A) does not have a particular solution of the form $$y_p=A\cos x+B\sin x$$.

32. Prove: If $$M$$, $$N$$ are constants (not both zero) and $$\omega>0$$, the constant coefficient equation

$ay''+by'+cy=M\cos\omega x+N\sin\omega x \tag{A}$

has a particular solution that’s a linear combination of $$\cos\omega x$$ and $$\sin\omega x$$ if and only if the left side of (A) is not of the form $$a(y''+\omega^2y)$$, so that $$\cos\omega x$$ and $$\sin\omega x$$ are solutions of the complementary equation.

## Q5.3.9

In Exercises 5.3.33-5.3.38 refer to the cited exercises and use the principal of superposition to find a particular solution. Then find the general solution.

33. $$y''+5y'-6y=22+18x-18x^2+6e^{3x}$$ (See Exercises 5.3.1 and 5.3.16.)

34. $$y''-4y'+5y=1+5x+e^{2x}$$ (See Exercises 5.3.2 and 5.3.17.)

35. $$y''+8y'+7y=-8-x+24x^2+7x^3+10e^{-2x}$$ (See Exercises 5.3.3 and 5.3.18.)

36. $$y''-4y'+4y=2+8x-4x^2+e^{x}$$ (See Exercises 5.3.4 and 5.3.19.)

37. $$y''+2y'+10y=4+26x+6x^2+10x^3+e^{x/2}$$ (See Exercises 5.3.5 and 5.3.20.)

38. $$y''+6y'+10y=22+20x+e^{-3x}$$ (See Exercises 5.3.6 and 5.3.21.)

## Q5.3.10

39. Prove: If $$y_{p_1}$$ is a particular solution of

$P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)$

on $$(a,b)$$ and $$y_{p_2}$$ is a particular solution of

$P_0(x)y''+P_1(x)y'+P_2(x)y=F_2(x)$

on $$(a,b)$$, then $$y_p=y_{p_1}+y_{p_2}$$ is a solution of

$P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)+F_2(x)$

on $$(a,b)$$.

40. Suppose $$p$$, $$q$$, and $$f$$ are continuous on $$(a,b)$$. Let $$y_1$$, $$y_2$$, and $$y_p$$ be twice differentiable on $$(a,b)$$, such that $$y=c_1y_1+c_2y_2+y_p$$ is a solution of

$y''+p(x)y'+q(x)y=f$

on $$(a,b)$$ for every choice of the constants $$c_1,c_2$$. Show that $$y_1$$ and $$y_2$$ are solutions of the complementary equation on $$(a,b)$$.

This page titled 5.3.1: Nonhomgeneous Linear Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.