# Thread: Integral Eqn - laplace tranforms

1. ## Integral Eqn - laplace tranforms

I am suppose to solve the following equation using laplace transforms and have started but am both unsure if i'm going about it the right way and where to go next,

$

y(t) + \int_{0}^{t} e^{2(t - T)} . y(T) . dT = e^{2t} - t

$

where T is supposed to be tau.

i re-arranged the equation to have y(t) = the rest , then used convolution theorem to say;

$
y(t) = e^{2t} - t -$
(y * $e^{2}$)(t)

where (y * $e^{2}$)(t) is the convolution

i then took la place transforms to give;
$
L(y(t)) = \frac{1}{s-2} - \frac{1}{s^{2}} - (L(y(t)) . \frac{1}{s-2}

$

From here i simplify but end up with terms i can't take the inverse laplace transform of in order to solve, so am thinking i must ahve gone wrong up to this point?
Can anyone help here?

Cheers,

2. Use y-hat for the transform and when I take the transform of the equation and simplify the convolution, I get: $\widehat{y}+\frac{1}{s-2}\widehat{y}=\frac{1}{s-2}-\frac{1}{s^2}$. Solving for yhat, I get: $\widehat{y}=\frac{2}{s-1}-\frac{2}{s^2}-\frac{1}{s}$. Now, you can take the inverse transform of that right?

3. Yes thanks!

My problem was in simplifying to something i could take the inverse transform of, but finally got it there.

Cheers.