Do it as two parts- first solve the "corresponding homogeneous equation" which is y"+ w^2y= 0. That has characteristic equation 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.