Find all solutions for the following ODE:
$\displaystyle y''+w^2y=Acos(wx), A /= 0 , w>0$
I figured it might be easier to solve as:
$\displaystyle
y''+w^2y-Acos(wx)=0$
but that still has me somewhat confused.
Do it as two parts- first solve the "corresponding homogeneous equation" which is y"+ w^2y= 0. That has characteristic equation $\displaystyle r^2+ w^2= 0$ which has roots iw and -iw. Do you know how to construct the general solution to the homogeneous differential equation from that?
Now, search for a single solution to the entire equation. Normally, I would suggest something of the form y= A cos(wx)+ B sin(wx) but after you have solved the corresponding homogeneous equation, you may see why you should try y= A x cos(wx)+ B x sin(wx) instead. Find y" for that, put into the equation and solve for A and B.
The general solution to the entire equation is the sum of the general solution to the corresponding homogeneous equation and any single solution to the entire equation.