4.5E: Nonhomgeneous Linear Equations (Exercises)
- Page ID
- 43297
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Q4.5.1
In Exercises 4.5.1-4.5.12 find a particular solution. 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. \(y''+6y'+10y=22+20x, \quad y(0)=2,\; y'(0)=-2\)
7. \(y''+5y'-6y=6e^{3x}\)
8. \(y''-4y'+5y=e^{2x}\)
9. \(y''+8y'+7y=10e^{-2x}, \quad y(0)=-2,\; y'(0)=10\)
10. \(y''-4y'+4y=e^{x}, \quad y(0)=2,\quad y'(0)=0\)
11. \(y''+2y'+10y=e^{x/2}\)
12. \(y''+6y'+10y=e^{-3x}\)
Q4.5.2
13. Show that \[y''+y'=1+2x+x^2; \tag{A}\] will not yield a particular solution of the form \(y_p=A+Bx+Cx^2\), where \(A\), \(B\), and \(C\) are constants.
14. Show that \[y''-7y'+12y=5e^{4x}; \tag{A}\] will not yield a particular solution of the form \(y_p=Ae^{4x}\).
15. 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.
Q4.5.3
In Exercises 4.5.16-4.5.21 find a particular solution. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.
16. \(y''-8y'+16y=23\cos x-7\sin x\)
17. \(y''+y'=-8\cos2x+6\sin2x\)
18. \(y''-2y'+3y=-6\cos3x+6\sin3x\)
19. \(y''+6y'+13y=18\cos x+6\sin x\)
20. \(y''+7y'+12y=-2\cos2x+36\sin2x, \quad y(0)=-3,\quad y'(0)=3\)
21. \(y''-6y'+9y=18\cos3x+18\sin3x, \quad y(0)=2,\quad y'(0)=2\)
Q4.5.4
In Exercises 4.5.22-5.3.27 use the principal of superposition to find a particular solution. Then find the general solution.
22. \(y''+5y'-6y=22+18x-18x^2+6e^{3x}\)
23. \(y''-4y'+5y=1+5x+e^{2x}\)
24. \(y''+8y'+7y=-8-x+24x^2+7x^3+10e^{-2x}\)
25. \(y''-4y'+4y=2+8x-4x^2+e^{x}\)
26. \(y''+2y'+10y=4+26x+6x^2+10x^3+e^{x/2}\)
27. \(y''+6y'+10y=22+20x+e^{-3x}\)
Q4.5.5
28. 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 \((a,b)\).29. 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 \((a,b)\).