# Homework Help: Using Laplace transforms to solve differential equations - with a twist!

1. Sep 21, 2014

### Laura W

I've been given this:
x''+ x = 4δ(t-2π)

With initial conditions of x(0) = 1 and x'(0) = 0, find x(t) using Laplace transforms.

I can easily get it to this:

4(sin(t-2π)u(t-2π))

But the question says "express your final solution without use of the unit step function". This is where I get confused as I'm not quite sure as to how to do that. Will it just be 4sin(t)? Considering the sin function repeats at 2π.

2. Sep 22, 2014

### LCKurtz

In general if you have $x=f(t)u(t-a)$ you would write a two piece function since $u(t-a) = 0$ if $t<a$ and $u(t-a) = 1$ if $t>a$. So you would say$$x =\left\{\begin{array}{l} 0,~t<a\\ f(t),~t>a \end{array}\right.$$Does that help?

3. Sep 22, 2014

### Laura W

That's what I was thinking but I was wondering if there was another way of transforming it at all? Seems to simple of an answer for this specific lecturer who wrote the question.

4. Sep 22, 2014

### vela

Staff Emeritus
Your answer isn't correct. Did you forget to incorporate the initial conditions?