# Solve the initial value problem below using the method of Laplace transforms y"-35y=144t-36^{-6t} y(0)=0 y'(0)=47

Question
Laplace transform
Solve the initial value problem below using the method of Laplace transforms
$$y"-35y=144t-36^{-6t}$$
$$y(0)=0$$
$$y'(0)=47$$

2020-11-18
Solution:
The differential equation is
$$y"-35y=144t-36^{-6t}$$
$$y(0)=0$$
$$y'(0)=47$$
Apply Laplace transforms:
$$L\left\{y"-35y\right\}=L\left\{144t-36^{-6t}\right\}$$
$$s^2Y(s)-sy(0)-y'(0)-36Y(s)=\frac{144}{s^2}-\frac{36}{s+6}$$
$$(s^2-36)Y(s)-47=\frac{144}{s^2}-\frac{36}{s+6}$$
$$Y(s)=\frac{144}{s^2(s^2-36)}-\frac{36}{(s+6)(s^2-36)}+\frac{47}{(s^2-36)}$$
Decompose each term:
$$\frac{144}{s^2(s^2-36)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{s+6}+\frac{D}{s-6}$$
$$=\frac{As(s^2-36)}{s^2(s^2-36)}+\frac{B(s^2-36)}{s^2(s^2-36)}+\frac{Cs^2(s-6)}{s^2(s^2-36)}+\frac{Ds^2(s+6)}{s^2(s^2-36)}$$
$$=\frac{(A+C+D)s^3+(B-6C+6D)s^2-36As-36B}{s^2(s^2-36)}$$
$$A=0 , B=-4 , C=-\frac{1}{3} ,\text{and } D=\frac{1}{3}$$
Step 2
$$\frac{144}{s^2(s^2-36)}=\frac{-4}{s^2}-\frac{1}{3(s+6)}+\frac{1}{3(s-6)}$$
$$\frac{36}{(s+6)(s^2-36)}=\frac{A}{s+6}+\frac{B}{(s+6)^2}+\frac{C}{(s-6)}$$
$$=\frac{A(s^2-36)+B(s-6)+C(s+6)^2}{(s+6)(s^2-36)}$$
$$A=\frac{1}{4} , B=3 , C=-\frac{1}{4}$$
$$\frac{36}{(s+6)(s^2-36)}=\frac{1}{4(s+6)}+\frac{3}{(s+6)^2}-\frac{1}{4(s-6)}$$
$$\frac{47}{(s^2-36)}=\frac{A}{s+6}+\frac{B}{s-6}$$
$$A=-\frac{47}{12} ,\ B=\frac{47}{12}$$
$$\text{Hence, } \frac{47}{(s^2-36)}=-\frac{47}{12(s+6)}+\frac{47}{12(s-6)}$$
Conclusion:
Hence, we have
$$Y(s)=\frac{-4}{s^2}-\frac{1}{3(s+6)}+\frac{1}{3(s-6)}-\frac{1}{4(s+6)}+\frac{3}{(s+6)^2}-\frac{1}{4(s-6)}-\frac{47}{12(s+6)}+\frac{47}{12(s-6)}$$
$$Y(s)=\frac{-4}{s^2}-\frac{9}{2(s+6)}+\frac{4}{(s-6)}+\frac{3}{(s+6)^2}$$
$$\text{Apply inverse Laplace transform: }$$
$$L^{-1}\left\{Y(s)\right\}=-4L^{-1}\left\{\frac{1}{s^2}\right\}-\frac{9}{2}L^{-1}\left\{\frac{1}{(s+6)}\right\}+4L^{-1}\left\{\frac{1}{(s-6)}\right\}+3L^{-1}\left\{\frac{1}{(s+6)^2}\right\}$$
$$y(t)=-4t-\frac{9}{2}e^{-6t}+4e^{6t}+3te^{-6t}$$
$$\text{Therefore, the solution is } y(t)=-4t-\frac{9}{2}e^{-6t}+4e^{6t}+3te^{-6t}$$

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