Laplace of $x^2\frac{d^2y}{dx^2}$

Question

How does one evaluate the Laplace of functions like $t^2\frac{d^2y}{dt^2}$ ?

I wanted to solve a differential equation using Laplace Transform resembling: $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 5$$

MATLAB Provides me the answer as :

C6*cos(log(t)) + C5*sin(log(t)) + 5

Can someone give me a derivation for this?

Answer 1

Use ${\cal L} \{ t^n f(t)\}=(-1)^n {d^n\over ds^n} F(s)$ and ${\cal L} \{ f''(t)\}=s^2F(s)-sf(0)-f'(0)$.

Answer 2

Take a new variable $x=e^t$.Then,

$$\frac{d}{dx}=\frac{d}{dt}\frac{dt}{dx}=\frac{d}{dt}e^{-t}$$

and

$$\frac{d^2}{dx^2}=\frac{d^2}{dt^2}e^{-2t}$$

and so your equation just becomes

$$\frac{d^2y}{dt^2}+\frac{dy}{dt}+y=5$$

that can be solved by combinations of sine and cosine.