# Homework Help: Using Laplace Transforms to Solve DE

1. Dec 7, 2008

### Ithryndil

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

$$y'' - 4y = e^{-t} , y(0) = 1, y'(0) = 0$$

2. Relevant equations
L{y(t)} = Y(s)
L{y'(t)} = sY(s) - y(0)
L{y''(t)} = s^2Y(s) - sy(0) - y(0)

3. The attempt at a solution

Ok, so I plugged the Laplace transforms for y'' and y into the equation as well as for e^(-t) and got:

$$Y(s) = \frac{1}{s(s-4)(s+1)} - \frac{1}{(s-4)} - \frac{4}{4(s+1)}$$

From that point on I would need to perform a partial fraction decomposition on the first term and last term, the middle term is ok. Doing so I get:

$$Y(s) = \frac{-1}{4s} + \frac{-1}{20(s-4)} + \frac{1}{5(s+1)}- \frac{1}{(s-1)} + \frac{1}{s} - \frac{1}{(s-4)}$$

However, the book is getting answers with e^(2t). I am not seeing how anything I have in that last line will yield that, so I assume I am doing something wrong.

2. Dec 7, 2008

### rock.freak667

Check over the transform of the left side. I believe you are supposed to get

s2Y(s)-sY(0)-Y'(0)-4Y(s)

3. Dec 7, 2008

### Ithryndil

...It's stupid mistakes like the one I made above that are frustrating. Thank you.