# Solving Diff EQ using a Laplace Transform

1. May 6, 2007

### G01

1. The problem statement, all variables and given/known data
Solve the initial value problem:

$$\frac{dy}{dt} + 2y = u_2(t)e^{-t}$$

y(0) = 3

Where $$u_2(t)$$ is a Heaviside Function with the discontinuity at t=2.

2. Relevant equations
The Laplace transform of a Heaviside function multiplied by another function:

$$L( u_a(t)f((t-a) ) = e^{-as}L(f(t-a))$$ Where L denotes the laplace tranform of a function.

3. The attempt at a solution

I know that in order to solve this equations using a laplace transform, I need to convert the RHS to the form of function in part 2. above. Once I do that I can take the Laplace Transform of both sides and then solve for L(y) and then y. I've been working at this for a while now, and I'm stuck on converting the RHS into a function whose transform I know. If I get this, then I can definitely do the rest of the problem. Any hints of converting this function into a workable form will be greatly appreciated.

2. May 6, 2007

### G01

Figured it out! Thanks anyway!