Science & engineering math: integro-differential equation

chatterbug219

1. The problem statement, all variables and given/known data

[itex]\int[/itex] y'(u)y(t-u)du = 24t³
The integral goes from t (top) to 0 (bottom)
With y(0) = 0

2. Relevant equations

I want to say it kind of looks like a convolution problem
[itex]\int[/itex] f(u)g(t-u)du
The integral goes from t (top) to 0 (bottom)

3. The attempt at a solution
I have no idea...

sunjin09

When you see a convolution in a homework problem, you should immediately think of a transform that reduces it to a multiplication. This transform can even handle the derivative easily...

chatterbug219

Well if it is convolution then it would just be
F(s)*G(s)
But I was more concerned about whether or not it actually was convolution. Because its y' and y, and those are both completely different, then it would be convolution then?
So I would need to take the Integral of y'(u) and y(t-u) then? And the integral of 24t³?

sunjin09

it is a convolution between to functions, i.e., y'(t) and y(t), their Laplace transform are related, because one is the derivative of the other. So there is only one F(s) to solve for, the other is simply related to this one.

chatterbug219

Okay, understandable...but how do I start the problem? I'm still confused about how to start...

Ray Vickson

What is the Laplace transform of a convolution?

RGV

chatterbug219

F(s)G(s)

HallsofIvy

So "start" by taking the Laplace transform of both sides!