ODE help, not sure what method to use

[Damascus Road]

I have the ODE:

$$y^{(4)}+2y''+y = 3 + cos(2x)$$

I believe I can use undetermined coefficients for the particular, but I'm not sure and it isn't working well for me so far, and the homogeneous looks nasty and I'm not sure what to attempt with.

Thanks!

 

[arildno]

Now, let's take the homogenous trial solution, $$y_{h}(x)=Ce^{kx}$$

Thus, the characteristic equation can be written as:
$$k^{4}+2k^{2}+1=0\to(k^{2}+1)^{2}=0\to{k}^{2}+1=0$$
This ought to be readily solvable for two of the roots.

Don't give up even before you had tried!