How to Solve a Non-Homogeneous Laplace Equation?

[reece]

Hi, I can solve homogeneous laplace fine, but with RHS i get stuck half way through.

Q:
y' +3y = 8e^{t}
y(0) = 2

Working as if it was homogeneous..

sY(s) - 2 + 3Y(s) = 8 . \frac{1}{s-1}
Y(s) (s+3) - 2 = 8 . \frac{1}{s-1}

I think the next step is
Y(s) = \frac{2}{s+3} + \frac{8}{s-1}
and then do partial fractions but i don't think it leads me to where I need to be. I think i need to make it into a heaviside ??

Any help would be great. thanks

 

[coomast]

Hello reece,
It took me a few minutes to figure out what was going on. Your method is OK, except that you should rewrite the final line as:
(s+3)Y(s)=\frac{8}{s-1}+2=2\cdot \frac{s+3}{s-1}
From which:
Y(s)=\frac{2}{s-1}
And thus the final solution is:
y(t)=2e^t

I expected two exponentials and therefore it is interesting to solve it without using Laplace. The solution is then
y(t)=A e^{-3t}+2e^t
After applying the boundary condition you get A=0.

It is a strange equation because the exponential is unbounded for large t.