
# 17.2E: Exercises for Section 17.2


Solve the following equations using the method of undetermined coefficients.

1.  $$2y''−5y′−12y=6$$

2.  $$3y''+y′−4y=8$$

$$y=c_1e^{−4x/3}+c_2e^x−2$$

3.  $$y''−6y′+5y=e^{−x}$$

4.  $$y''+16y=e^{−2x}$$

$$y=c_1 \cos4x+c_2 \sin 4x+\frac{1}{20}e^{−2x}$$

5.  $$y″−4y=x^2+1$$

6.  $$y″−4y′+4y=8x^2+4x$$

$$y=c_1e^{2x}+c_2xe^{2x}+2x^2+5x$$

7.  $$y″−2y′−3y= \sin 2x$$

8.  $$y″+2y′+y= \sin x+ \cos x$$

$$y=c_1e^{−x}+c_2xe^{−x}+\frac{1}{2} \sin x−\frac{1}{2} \cos x$$

9.  $$y″+9y=e^x \cos x$$

10.  $$y″+y=3 \sin 2x+x \cos 2x$$

$$y=c_1 \cos x+ c_2 \sin x−\frac{1}{3}x \cos 2x−\frac{5}{9} \sin 2x$$

11.  $$y″+3y′−28y=10e^{4x}$$

12.  $$y″+10y′+25y=xe^{−5x}+4$$

$$y=c_1e^{−5x}+c_2xe^{−5x}+\frac{1}{6}x^3e^{−5x}+\frac{4}{25}$$

In exercises 13 - 18,

1. Write the form for the particular solution $$y_p(x)$$ for the method of undetermined coefficients.
2. [T] Use a computer algebra system to find a particular solution to the given equation.

13.  $$y″−y′−y=x+e^{−x}$$

14.  $$y″−3y=x^2−4x+11$$

a. $$y_p(x)=Ax^2+Bx+C$$

b. $$y_p(x)=−\frac{1}{3}x^2+\frac{4}{3}x−\frac{35}{9}$$

15.  $$y''−y′−4y=e^x \cos 3x$$

16.  $$2y″−y′+y=(x^2−5x)e^{−x}$$

a. $$y_p(x)=(Ax^2+Bx+C)e^{−x}$$

b. $$y_p(x)=(\frac{1}{4}x^2−\frac{5}{8}x−\frac{33}{32})e^{−x}$$

17.  $$4y″+5y′−2y=e^{2x}+x \sin x$$

18.  $$y''−y′−2y=x^2e^x \sin x$$

a. $$y_p(x)=(Ax^2+Bx+C)e^x \cos x+(Dx^2+Ex+F)e^x \sin x$$

b. $$y_p(x)=(−\frac{1}{10}x^2−\frac{11}{25}x−\frac{27}{250})e^x \cos x +(−\frac{3}{10}x^2+\frac{2}{25}x+\frac{39}{250})e^x \sin x$$

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

19.  $$y″+3y′−4y=2e^x$$

20.  $$y''+2y′=e^{3x}$$

$$y=c_1+c_2e^{−2x}+\frac{1}{15}e^{3x}$$

21.  $$y''+6y′+9y=e^{−x}$$

22.  $$y''+2y′−8y=6e^{2x}$$

$$y=c_1e^{2x}+c_2e^{−4x}+xe^{2x}$$

Solve the differential equation using the method of variation of parameters.

23.  $$4y″+y=2 \sin x$$

24.  $$y″−9y=8x$$

$$y=c_1e^{3x}+c_2e^{−3x}−\frac{8x}{9}$$

25.  $$y″+y= \sec x, \quad 0<x<π/2$$

26.  $$y″+4y=3 \csc 2x, \quad 0<x<π/2$$

$$y=c_1 \cos 2x+c_2 \sin 2x−\frac{3}{2} x \cos 2x+\frac{3}{4} \sin 2x \ln ( \sin 2x)$$

Find the unique solution satisfying the differential equation and the initial conditions given, where $$y_p(x)$$ is the particular solution.

27.  $$y″−2y′+y=12e^x,\quad y_p(x)=6x^2e^x, \; y(0)=6, \; y′(0)=0$$

28.  $$y''−7y′=4xe^{7x},\quad y_p(x)=\frac{2}{7}x^2e^{7x}−\frac{4}{49}xe^{7x}, \; y(0)=−1, \; y'(0)=0$$

$$y=− \frac {347}{343}+ \frac {4}{343}e^{7x}+\frac{2}{7}x^2e^{7x}−\frac{4}{49}xe^{7x}$$

29.  $$y″+y= \cos x−4 \sin x, \quad y_p(x)=2x \cos x+\frac{1}{2} x \sin x, \; y(0)=8, \; y′(0)=−4$$

30.  $$y″−5y′=e^{5x}+8e^{−5x}, \quad y_p(x)=\frac{1}{5}xe^{5x}+\frac{4}{25}e^{−5x}, \; y(0)=−2, \; y′(0)=0$$

$$y=−\frac{57}{25}+\frac{3}{25}e^{5x}+\frac{1}{5}xe^{5x}+\frac{4}{25}e^{−5x}$$

In problems 31 - 32, two linearly independent solutions—$$y_1$$ and $$y_2$$—are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume $$x>0$$ in each exercise.

31.  $$x^2y″+2xy′−2y=3x, \quad y_1(x)=x, \; y2(x)=x^{−2}$$

32.  $$x^2y''−2y=10x^2−1,\quad y_1(x)=x^2, \; y_2(x)=x^{−1}$$

$$y_p=\frac{1}{2}+\frac{10}{3}x^2 \ln x$$