1. How do you find a particular solution to a second order DE with constant coefficients using linear operators? Does the inputs have to be either e, sin, cos or always some combination of them to use linear operators?

Also, is there a general method for solving second order homogeneous DE with non-constant coefficients that are NOT in the Cauchy-Euler form and that does not use power series? Thanks !

2. Originally Posted by Zero266
How do you find a particular solution to a second order DE with constant coefficients using linear operators? Does the inputs have to be either e, sin, cos or always some combination of them to use linear operators?
See here.

Also, is there a general method for solving second order homogeneous DE with non-constant coefficients that are NOT in the Cauchy-Euler form and that does not use power series? Thanks !
I think Laplace Transforms can be used to take care of these nasties IFF there are initial conditions given. See here.

Otherwise I'm not quite sure of any other technique besides using power series...

I would consider conducting a Google search

3. Originally Posted by Chris L T521
I think Laplace Transforms can be used to take care of these nasties IFF there are initial conditions given. See here.

Otherwise I'm not quite sure of any other technique besides using power series...

I would consider conducting a Google search
Just to add a little bit, using the annilator approach is usually limited to nonhomogeneous terms like you said $e^{ax}, x^n, \sin bx,\; \text{and}\; \cos b x$ and combinations thereof. But for more complicated nonhomogeneous terms, it seldom works. A classic example is in the following

$y'' + y = \sec x$

How does one annilate $\sec x$ term?

As for the Laplace transform method, often one obtains an ODE harder than the one you started with. With simple power law coefficients, it might be manageable (see Chris L T521 link). I might add that if one was clever enough (or lucky enough) to guess one solution, then using $y = u y_l$

where $y_l$ was your guessed solution, would reduce your problem to first order. I might also add that every ODE of the form

$y'' + p(x) y' + q(x) y = 0$

can be reduced to a Ricatti equation under the substitution

$y' = u y$.