# Thread: Laplace Transform of a DE

1. ## Laplace Transform of a DE

First time solving DEs this way so bare with me.

Consider $\ddot{y} + 2 \dot{y} + 5y = u(t)$ with $y_0 = 1$ and $\dot{y}_0 = -1$.

Using the Laplace Transform of both sides we get:

$(s^2 Y(s) - sy_0 - \dot{y}_0) + 2(s Y(s) - y_0) + 5Y(s) - U(s)$

Isolating $Y(s)$ and using the initial conditions, we get:

$Y(s) = \frac{U(s) + s + 1}{s^2 + 2s + 5}$

Now in the answers I've got it says let $u(t) = 2t-1$ and they then proceed to do the Laplace Transform of that and substitute it in.

Just wondering why they've chosen $u(t) = 2t-1$, is this something I've missed?

2. ## Re: Laplace Transform of a DE

Actually it doesn't matter, turns out this is just following on from another example!!