# Math Help - Convolution of a DE

1. ## Convolution of a DE

Hi,

I don't even know where or how to start this problem and really need help.

The problem is:
Consider the differential equation 
Find a function
 , so that the particular solutions is given by the convolution 

I don't know what the question is really asking and how to approach it.

Thanks!

2. ## Re: Convolution of a DE

Originally Posted by JC05
Hi,

I don't even know where or how to start this problem and really need help.

The problem is:
Consider the differential equation 
Find a function
 , so that the particular solutions is given by the convolution 
[SIZE=2]
I don't know what the question is really asking and how to approach it.
Do you know Laplace transforms and how to use them to solve a DE?

taking Laplace transforms of both sides

$(s^2+4s+13)X(s) = F(s)$

$X(s)=\dfrac {F(s)}{s^2+4s+13}=\Theta(s) F(s)$ where

$\Theta(s)=\dfrac 1 {s^2+4s+13}$

multiplication in the $s$ domain corresponds to convolution in the $t$ domain so

$\Theta(s)F(s) \Longleftrightarrow \theta(t) * f(t)$

and thus

$\theta(t) = \mathscr{L}^{-1} \{ \Theta(s) \} = \mathscr{L}^{-1} \left\{\dfrac 1 {s^2+4s+13} \right\}$