Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

Question

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$

$(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.

Answer 1

Should that be $e^{3t}$ instead of $e^{3}t$?

You have a second order linear ODE, so the solution will be the homogenous solution plus a particular solution. The homogenous solution should be easy to work out. For the particular solution, use part (a). Specifically assume the particular solution $f(t)$ satisfies $f(t)=e^{r_1t}v''(t)$. Then:

$0=L(f)-t^{3/2}e^{3t}=e^{r_1 t}v''(t)-t^{3/2}e^{3t}$.

So $v''(t)=t^{3/2}e^{(3-r_1)t}$. Can you finish it from here?