# Math Help - HELP LAPLACE TRANSFORM!!!!

1. ## HELP LAPLACE TRANSFORM!!!!

Hey. My exam is in a few hours so it'd be cool if someone could show me this. I can guarantee you it won't hinder my understanding by just "giving me the answer"... nothing would benefit me more at this point haha.

Anyway. I don't even know if it's possible but can you find a Laplace transform of this in terms of $Y(s)$:

$\mathcal L (y(t).sin(\omega t))$

It's part of a bigger question where I have to find the transfer function of the differential equation:

$\ddot{x} + \zeta \omega_n \dot{x} + \omega_n^2 x = y(t)sin(\omega t)$

Assuming zero initial conditions: Hence my LHS is:

$(s^2 + \zeta \omega_n s + \omega_n^2)X(s) = \mathcal L (y(t).sin(\omega t) )$

The transfer function is given by the ratio of the frequency-domain input to the frequency-domain output. The input being $Y(s)$, and the output being $X(s)$.

In short I need to get $Y(s)$ on the RHS of that equation using the laplace transform integral I'd assume, and then manipulate the equation to get $\frac{X(s)}{Y(s)}$

(NOTE: $\omega_n, \omega, \zeta$ are all constants.[/tex]

Many thanks.

2. I saw that yesterday but I could only get it in terms of $Y(s-i\omega)$ since:

$L\left\{y(t)\sin(\omega t)\right\}=\text{Im}\left( L\left\{e^{i\omega t} y(t)\right\}\right)=\text{Im}\left\{Y(s-i\omega)\right\}$