Integral equation to solve with La Place transformation

[Funzies]

Homework Statement

$$y(t)+2\int_0^tcos(t-\tau)y(\tau)d\tau = 9e^{2t}$$
Solve this integral equation by using the La Place transform.

Homework Equations

None

The Attempt at a Solution

I tried differentiation the whole equation:
$$y'(t) + 2cos(t-\tau)y(\tau) = 18e^{2t}$$ with boundaries 0 and t
$$y'(t) + 2y(t) +cos(t)y(0) = 18e^{2t}$$
The exercise didn't give a value for y(0), so I assumed it to be zero.
$$y'(t) + 2y(t) = 18e^{2t} - cos(t)$$
By using the La Place transform:
$$Sy(s) + 2y(s) = 18/(s-2) - s/(s^2+1)$$
$$y(s) = \18/(s-2)(s+2) - s/(s^2+1)(s+2)$$
This is were I started doubting my approach. I tried to rewrite it in 3 fractions and wanted to use the inverse la Place transform on these fractions. I got some very nasty numbers though and as the answer is very straight-forward I started doubting my approach. Can anyone comment on the correctness of my approach?

 

[stevenb]

I don't think your approach is the best way, and even if it is viable, I think I see a few mistakes in there, looking quickly.

I recommend dealing with the equation directly. Note that the integral is a convolution integral, which is different than a standard integral. Now think about the property that convolution in the time domain is equivalent to multiplication in the Laplace domain.