5.4.1: Nonhomgeneous Linear Equations (Exercises)
- Page ID
- 103504
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
In Exercises 1-18 verify that the given \(y_p\) is particular solution of the nonhomogeneous differential equation given and form the general solution or, where initial values are given, solve the initial value problem.
1. \(y''+5y'-6y=22+18x-18x^2; \quad y_{p}=-1+2x+3x^{2}\)
2. \(y''-4y'+5y=1+5x; \quad y_{p}=1+x\)
3. \(y''+8y'+7y=-8-x+24x^2+7x^3; \quad y_{p}=-x+x^{3} \)
4. \(y''-4y'+4y=2+8x-4x^2; \quad y_{p}= 1 − x^{2} \)
5. \(y''+2y'+10y=4+26x+6x^2+10x^3, \quad y(0)=2, \quad y'(0)=9 1 − x^{2}; \quad y_{p}= 2x + x^{3} \)
6. \(y''+6y'+10y=22+20x, \quad y(0)=2,\; y'(0)=-2; \quad y_{p}= 1 + 2x \)
7. \(y''+5y'-6y=6e^{3x}; \quad y_{p}= \frac{e^{3x}}{3} \)
8. \(y''-4y'+5y=e^{2x}; \quad y_{p}= e^{2x} \)
9. \(y''+8y'+7y=10e^{-2x}, \quad y(0)=-2,\; y'(0)=10; \quad y_{p}= −2e^{−2x} \)
10. \(y''-4y'+4y=e^{x}, \quad y(0)=2,\quad y'(0)=0; \quad y_{p}= e^{x} \)
11. \(y''+2y'+10y=e^{x/2}; \quad y_{p}= \frac{4}{45}e^{x/2} \)
12. \(y''+6y'+10y=e^{-3x}; \quad y_{p}= e^{−3x} \)
13. \(y''-8y'+16y=23\cos x-7\sin x; \quad y_{p}= \cos x − \sin x \)
14. \(y''+y'=-8\cos2x+6\sin2x; \quad y_{p}= \cos 2x − 2 \sin 2x \)
15. \(y''-2y'+3y=-6\cos3x+6\sin3x; \quad y_{p}= \cos 3x \)
16. \(y''+6y'+13y=18\cos x+6\sin x; \quad y_{p}= \cos x + \sin x \)
17. \(y''+7y'+12y=-2\cos2x+36\sin2x, \quad y(0)=-3,\quad y'(0)=3; \quad y_{p}= −2 \cos 2x + \sin 2x \)
18. \(y''-6y'+9y=18\cos3x+18\sin3x, \quad y(0)=2,\quad y'(0)=2; \quad y_{p}= \cos 3x − \sin 3x \)
In Exercises 19-24 refer to the cited exercises and use the principle of superposition to find a particular solution. Then find the general solution.
19. \(y''+5y'-6y=22+18x-18x^2+6e^{3x}\) (See Exercises 5.4.1 and 5.4.7.)
20. \(y''-4y'+5y=1+5x+e^{2x}\) (See Exercises 5.4.2 and 5.4.8)
21. \(y''+8y'+7y=-8-x+24x^2+7x^3+10e^{-2x}\) (See Exercises 5.4.3 and 5.4.9.)
22. \(y''-4y'+4y=2+8x-4x^2+e^{x}\) (See Exercises 5.4.4 and 5.4.10)
23. \(y''+2y'+10y=4+26x+6x^2+10x^3+e^{x/2}\) (See Exercises 5.3.5 and 5.4.11.)
24. \(y''+6y'+10y=22+20x+e^{-3x}\) (See Exercises 5.4.6 and 5.4.12.)
25. Prove: If \(y_{p_1}\) is a particular solution of
\[P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)\nonumber\]
on \((a,b)\) and \(y_{p_2}\) is a particular solution of
\[P_0(x)y''+P_1(x)y'+P_2(x)y=F_2(x)\nonumber\]
on \((a,b)\), then \(y_p=y_{p_1}+y_{p_2}\) is a solution of
\[P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)+F_2(x)\nonumber\]
on \((a,b)\).
26. 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
\[y''+p(x)y'+q(x)y=f\nonumber\]
on \((a,b)\) for every choice of the constants \(c_1,c_2\). Prove that \(y_1\) and \(y_2\) are solutions of the homogeneous equation on \((a,b)\).