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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?

2 Answers 2

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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).

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    I am just fiddling around. I don't really need to solve the equation using LT but was wondering if I could.2012-02-07
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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.

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    Yes, you can use Laplace on the last equation. But, I think that the hint by David could be more helpful.2012-02-07