Answer to Question #312153 in Differential Equations for haru

Question

8. Solve x

x^2y′′− 2xy′+ 2y = x^4sin(4 log x)

Answer 1

$x=e^u$

$a_0\lambda(\lambda-1)(\lambda-2)...(\lambda-n+1)...a_{n-2}\lambda(\lambda-1)+a_{n-1}\lambda+a_n=0$

$(\lambda-1)\lambda-2\lambda+2=0$

$\lambda^2-3\lambda+2=0$

$\lambda_1=2$

$\lambda_2=1$

$y=Ce^u+C_1e^{2u}$

$y_i=u^se^{au}(R_m(u)cos(bu)+T_m(u)sin(bu))$

s=0 if a+bi is not a root or s=k if it is.

a+bi=4i+4 s=0

$y_0=e^{4u}(Bsin(4u)+Acos(4u))$

$y'_0(u)=(4B-4A)e^{4u}sin(4u)+(4B+4A)e^{4u}cos(4u)$

$y''_0=32Be^{4u}cos(4u)-32Ae^{4u}sin(4u)$

$(-10B-20A)e^{4u}sin(4u)+(20B-10A)e^{4u}cos(4u)=e^{4u}sin(4u)$

$A=\frac{-1}{25}$

$B=\frac{-1}{50}$

$y_0=e^{4u}(\frac{-sin(4u)}{50}-\frac{cos(4u)}{25})$

$y=e^{4u}(\frac{-sin(4u)}{50}-\frac{cos(4u)}{25})+Ce^u+C_1e^{2u}$