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Mathematics LibreTexts

17.2E: Exercises for Section 17.2

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    72450
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    Solve the following equations using the method of undetermined coefficients.

    1.  \(2y''−5y′−12y=6\)

    2.  \(3y''+y′−4y=8\)

    Answer:
    \(y=c_1e^{−4x/3}+c_2e^x−2\)

    3.  \(y''−6y′+5y=e^{−x}\)

    4.  \(y''+16y=e^{−2x}\)

    Answer:
    \(y=c_1 \cos4x+c_2 \sin 4x+\frac{1}{20}e^{−2x}\)

    5.  \(y″−4y=x^2+1\)

    6.  \(y″−4y′+4y=8x^2+4x\)

    Answer:
    \(y=c_1e^{2x}+c_2xe^{2x}+2x^2+5x\)

    7.  \(y″−2y′−3y= \sin 2x\)

    8.  \(y″+2y′+y= \sin x+ \cos x\)

    Answer:
    \(y=c_1e^{−x}+c_2xe^{−x}+\frac{1}{2} \sin x−\frac{1}{2} \cos x\)

    9.  \(y″+9y=e^x \cos x\)

    10.  \(y″+y=3 \sin 2x+x \cos 2x\)

    Answer:
    \(y=c_1 \cos x+ c_2 \sin x−\frac{1}{3}x \cos 2x−\frac{5}{9} \sin 2x\)

    11.  \(y″+3y′−28y=10e^{4x}\)

    12.  \(y″+10y′+25y=xe^{−5x}+4\)

    Answer:
    \(y=c_1e^{−5x}+c_2xe^{−5x}+\frac{1}{6}x^3e^{−5x}+\frac{4}{25}\)

     

    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.  \(y″−3y=x^2−4x+11\)

    Answer:

    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.  \(2y″−y′+y=(x^2−5x)e^{−x}\)

    Answer:

    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.  \(y''−y′−2y=x^2e^x \sin x\)

    Answer:

    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.  \(y''+2y′=e^{3x}\)

    Answer:
    \(y=c_1+c_2e^{−2x}+\frac{1}{15}e^{3x}\)

    21.  \(y''+6y′+9y=e^{−x}\)

    22.  \(y''+2y′−8y=6e^{2x}\)

    Answer:
    \(y=c_1e^{2x}+c_2e^{−4x}+xe^{2x}\)

     

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

    23.  \(4y″+y=2 \sin x\)

    24.  \(y″−9y=8x\)

    Answer:
    \(y=c_1e^{3x}+c_2e^{−3x}−\frac{8x}{9}\)

    25.  \(y″+y= \sec x, \quad 0<x<π/2\)

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

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

     

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

    Answer:
    \(y=− \frac {347}{343}+ \frac {4}{343}e^{7x}+\frac{2}{7}x^2e^{7x}−\frac{4}{49}xe^{7x}\)

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

    Answer:
    \(y=−\frac{57}{25}+\frac{3}{25}e^{5x}+\frac{1}{5}xe^{5x}+\frac{4}{25}e^{−5x}\)

     

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

    Answer:
    \(y_p=\frac{1}{2}+\frac{10}{3}x^2 \ln x\)

     


    17.2E: Exercises for Section 17.2 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.