# 14.3E: The Method of Undetermined Coefficients II (Exercises)

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## Q5.5.1

In Exercises 5.5.1-5.5.17 find a particular solution.

1. $$y''+3y'+2y=7\cos x-\sin x$$

2. $$y''+3y'+y=(2-6x)\cos x-9\sin x$$

3. $$y''+2y'+y=e^x(6\cos x+17\sin x)$$

4. $$y''+3y'-2y=-e^{2x}(5\cos2x+9\sin2x)$$

5. $$y''-y'+y=e^x(2+x)\sin x$$

6. $$y''+3y'-2y=e^{-2x}\left[(4+20x)\cos 3x+(26-32x)\sin 3x\right]$$

7. $$y''+4y=-12\cos2x-4\sin2x$$

8. $$y''+y=(-4+8x)\cos x+(8-4x)\sin x$$

9. $$4y''+y=-4\cos x/2-8x\sin x/2$$

10. $$y''+2y'+2y=e^{-x}(8\cos x-6\sin x)$$

11. $$y''-2y'+5y=e^x\left[(6+8x)\cos 2x+(6-8x)\sin2x\right]$$

12. $$y''+2y'+y=8x^2\cos x-4x\sin x$$

13. $$y''+3y'+2y=(12+20x+10x^2)\cos x+8x\sin x$$

14. $$y''+3y'+2y=(1-x-4x^2)\cos2x-(1+7x+2x^2)\sin2x$$

15. $$y''-5y'+6y=-e^x\left[(4+6x-x^2)\cos x-(2-4x+3x^2)\sin x\right]$$

16. $$y''-2y'+y=-e^x\left[(3+4x-x^2)\cos x+(3-4x-x^2)\sin x\right]$$

17. $$y''-2y'+2y=e^x\left[(2-2x-6x^2)\cos x+(2-10x+6x^2)\sin x\right]$$

## Q5.5.2

In Exercises 5.5.18-5.5.21 find a particular solution and graph it.

18. $$y''+2y'+y=e^{-x}\left[(5-2x)\cos x-(3+3x)\sin x\right]$$

19. $$y''+9y=-6\cos 3x-12\sin 3x$$

20. $$y''+3y'+2y=(1-x-4x^2)\cos2x-(1+7x+2x^2)\sin2x$$

21. $$y''+4y'+3y=e^{-x}\left[(2+x+x^2)\cos x+(5+4x+2x^2)\sin x\right]$$

## Q5.5.3

In Exercises 5.5.22-5.5.26 solve the initial value problem.

22. $$y''-7y'+6y=-e^x(17\cos x-7\sin x), \quad y(0)=4,\; y'(0)=2$$

23. $$y''-2y'+2y=-e^x(6\cos x+4\sin x), \quad y(0)=1,\; y'(0)=4$$

24. $$y''+6y'+10y=-40e^x\sin x, \quad y(0)=2,\quad y'(0)=-3$$

25. $$y''-6y'+10y=-e^{3x}(6\cos x+4\sin x), \quad y(0)=2,\quad y'(0)=7$$

26. $$y''-3y'+2y=e^{3x}\left[21\cos x-(11+10x)\sin x\right], \; y(0)=0, \quad y'(0)=6$$

## Q5.5.4

In Exercises 5.5.27-5.5.32 use the principle of superposition to find a particular solution. Where indicated, solve the initial value problem.

27. $$y''-2y'-3y=4e^{3x}+e^x(\cos x-2\sin x)$$

28. $$y''+y=4\cos x-2\sin x+xe^x+e^{-x}$$

29. $$y''-3y'+2y=xe^x+2e^{2x}+\sin x$$

30. $$y''-2y'+2y=4xe^x\cos x+xe^{-x}+1+x^2$$

31. $$y''-4y'+4y=e^{2x}(1+x)+e^{2x}(\cos x-\sin x)+3e^{3x}+1+x$$

32. $$y''-4y'+4y=6e^{2x}+25\sin x, \quad y(0)=5,\; y'(0)=3$$

## Q5.5.5

In Exercises 5.5.33-5.5.35 solve the initial value problem and graph the solution.

33. $$y''+4y=-e^{-2x}\left[(4-7x)\cos x+(2-4x)\sin x\right], \; y(0)=3, \quad y'(0)=1$$

34. $$y''+4y'+4y=2\cos2x+3\sin2x+e^{-x}, \quad y(0)=-1,\; y'(0)=2$$

35. $$y''+4y=e^x(11+15x)+8\cos2x-12\sin2x, \quad y(0)=3,\; y'(0)=5$$

## Q5.5.6

36.

1. Verify that if $y_p=A(x)\cos\omega x+B(x)\sin\omega x \nonumber$ where $$A$$ and $$B$$ are twice differentiable, then \begin{aligned} y_p'&=(A'+\omega B)\cos\omega x+(B'-\omega A) \sin\omega x\quad \mbox{ and}\\[4pt] y_p''&=(A''+2\omega B'-\omega^2A)\cos\omega x +(B''-2\omega A'-\omega^2B)\sin\omega x.\end{aligned} \nonumber
2. Use the results of (a) to verify that \begin{aligned} ay_p''+by_p'+cy_p=&\left[(c-a\omega^2)A+b\omega B+2a\omega B'+bA'+aA''\right] \cos\omega x+\\[4pt] & \left[-b\omega A+(c-a\omega^2)B-2a\omega A'+bB'+aB''\right]\sin\omega x.\end{aligned} \nonumber
3. Use the results of (a) to verify that $y_p''+\omega^2 y_p=(A''+2\omega B')\cos\omega x+ (B''-2\omega A')\sin\omega x. \nonumber$
4. Prove Theorem 5.5.2.

37. Let $$a$$, $$b$$, $$c$$, and $$\omega$$ be constants, with $$a\ne0$$ and $$\omega>0$$, and let

$P(x)=p_0+p_1x+\cdots+p_kx^k \quad \text{and} \quad Q(x)=q_0+q_1x+\cdots+q_kx^k, \nonumber$

where at least one of the coefficients $$p_k$$, $$q_k$$ is nonzero, so $$k$$ is the larger of the degrees of $$P$$ and $$Q$$.

1. Show that if $$\cos\omega x$$ and $$\sin\omega x$$ are not solutions of the complementary equation $ay''+by'+cy=0, \nonumber$ then there are polynomials $A(x)=A_0+A_1x+\cdots+A_kx^k \quad \text{and} \quad B(x)=B_0+B_1x+\cdots+B_kx^k \tag{A}$ such that $\begin{array}{lcl} \quad (c-a\omega^2)A+b\omega B+2a\omega B'+bA'+aA''&=P\phantom{.}\\[4pt] -b\omega A+(c-a\omega^2)B-2a\omega A'+bB'+aB''&=Q, \end{array} \nonumber$ where $$(A_k,B_k)$$, $$(A_{k-1},B_{k-1})$$, …,$$(A_0,B_0)$$ can be computed successively by solving the systems $\begin{array}{lcl} \phantom{-}(c-a\omega^2)A_k+b\omega B_k&=p_k\phantom{.}\\[4pt] -b\omega A_k+(c-a\omega^2)B_k&=q_k, \end{array} \nonumber$ and, if $$1\le r\le k$$, $\begin{array}{lcl} \phantom{-}(c-a\omega^2)A_{k-r}+b\omega B_{k-r}&=p_{k-r}+\cdots\phantom{.}\\[4pt] -b\omega A_{k-r}+(c-a\omega^2)B_{k-r}&=q_{k-r}+\cdots, \end{array} \nonumber$ where the terms indicated by “$$\cdots$$” depend upon the previously computed coefficients with subscripts greater than $$k-r$$. Conclude from this and Exercise 5.5.36b that $y_p=A(x)\cos\omega x+B(x)\sin\omega x \tag{B}$ is a particular solution of $ay''+by'+cy=P(x)\cos\omega x+Q(x)\sin\omega x. \nonumber$
2. Conclude from Exercise 5.5.36c that the equation $a(y''+\omega^2 y)=P(x)\cos\omega x+Q(x)\sin\omega x \tag{C}$ does not have a solution of the form (B) with $$A$$ and $$B$$ as in (A). Then show that there are polynomials $A(x)=A_0x+A_1x^2+\cdots+A_kx^{k+1}\quad \text{and} \quad B(x)=B_0x+B_1x^2+\cdots+B_kx^{k+1} \nonumber$ such that $\begin{array}{rcl} a(A''+2\omega B')&=P\\[4pt] a(B''-2\omega A')&=Q, \end{array} \nonumber$ where the pairs $$(A_k,B_k)$$, $$(A_{k-1},B_{k-1})$$, …, $$(A_0,B_0)$$ can be computed successively as follows: \begin{aligned} A_k&=-{q_k\over2a\omega(k+1)}\\[4pt] B_k&=\phantom{-}{p_k\over2a\omega(k+1)},\end{aligned} \nonumber and, if $$k\ge 1$$, \begin{aligned} A_{k-j}&=-{1\over2\omega} \left[{q_{k-j}\over a(k-j+1)}-(k-j+2)B_{k-j+1}\right]\\[4pt] B_{k-j}&=\phantom{-}{1\over2\omega} \left[{p_{k-j}\over a(k-j+1)}-(k-j+2)A_{k-j+1}\right]\end{aligned} \nonumber for $$1\le j\le k$$. Conclude that (B) with this choice of the polynomials $$A$$ and $$B$$ is a particular solution of (C).

38. Show that Theorem 5.5.1 implies the next theorem:

##### Theorem 5.5E.1

Suppose $$\omega$$ is a positive number and $$P$$ and $$Q$$ are polynomials. Let $$k$$ be the larger of the degrees of $$P$$ and $$Q$$. Then the equation

$ay''+by'+cy=e^{\lambda x}\left(P(x)\cos \omega x+Q(x)\sin \omega x\right) \nonumber$

has a particular solution

$y_p=e^{\lambda x}\left(A(x)\cos\omega x+B(x)\sin\omega x\right), \tag{A}$

where

$A(x)=A_0+A_1x+\cdots+A_kx^k \quad \text{and} \quad B(x)=B_0+B_1x+\cdots+B_kx^k, \nonumber$

provided that $$e^{\lambda x}\cos\omega x$$ and $$e^{\lambda x}\sin\omega x$$ are not solutions of the complementary equation. The equation

$a\left[y''-2\lambda y'+(\lambda^2+\omega^2)y\right]= e^{\lambda x}\left(P(x)\cos \omega x+Q(x)\sin \omega x\right) \nonumber$

$$($$for which $$e^{\lambda x}\cos\omega x$$ and $$e^{\lambda x}\sin\omega x$$ are solutions of the complementary equation$$)$$ has a particular solution of the form (A), where

$A(x)=A_0x+A_1x^2+\cdots+A_kx^{k+1} \quad \text{and} \quad B(x)=B_0x+B_1x^2+\cdots+B_kx^{k+1}. \nonumber$

39. This exercise presents a method for evaluating the integral

$y=\int e^{\lambda x}\left(P(x)\cos \omega x+Q(x)\sin \omega x\right)\,dx \nonumber$

where $$\omega\ne0$$ and

$P(x)=p_0+p_1x+\cdots+p_kx^k,\quad Q(x)=q_0+q_1x+\cdots+q_kx^k. \nonumber$

1. Show that $$y=e^{\lambda x}u$$, where $u'+\lambda u=P(x)\cos \omega x+Q(x)\sin \omega x. \tag{A}$
2. Show that (A) has a particular solution of the form $u_p=A(x)\cos \omega x+B(x)\sin \omega x, \nonumber$ where $A(x)=A_0+A_1x+\cdots+A_kx^k,\quad B(x)=B_0+B_1x+\cdots+B_kx^k, \nonumber$ and the pairs of coefficients $$(A_k,B_k)$$, $$(A_{k-1},B_{k-1})$$, …,$$(A_0,B_0)$$ can be computed successively as the solutions of pairs of equations obtained by equating the coefficients of $$x^r\cos\omega x$$ and $$x^r\sin\omega x$$ for $$r=k$$, $$k-1$$, …, $$0$$.
3. Conclude that $\int e^{\lambda x}\left(P(x)\cos \omega x+Q(x)\sin \omega x\right)\,dx = e^{\lambda x}\left(A(x)\cos \omega x+B(x)\sin \omega x\right) +c, \nonumber$ where $$c$$ is a constant of integration.

40. Use the method of Exercise 5.5.39 to evaluate the integral.

1. $$\int x^{2}\cos x dx$$
2. $$\int x^{2} e^{x}\cos x dx$$
3. $$\int xe^{-x}\sin 2x dx$$
4. $$\int x^{2}e^{-x}\sin x dx$$
5. $$\int x^{3}e^{x}\sin x dx$$
6. $$\int e^{x}[x\cos x - (1+3x)\sin x ]dx$$
7. $$\int e^{-x}[(1+x^{2})\cos x +(1_x^{2})\sin x]dx$$

This page titled 14.3E: The Method of Undetermined Coefficients II (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.