Find all solutions to the differential equation:

$4y''+\beta^2 y = cos \beta x$

Where $\beta$ is a positive constant.

Starting with the homogeneous part :

$4r^2+\beta^2=0$

$r = \frac{i\beta}{2}$ since we only want the positive root.

Giving the solution:

$C\_1\cos(\frac{\beta x}{2})+C\_2\sin(\frac{\beta x}{2})$

Is this the correct way to start off? How would you find $y\_p$?