I know how to solve linear homogeneous ordinary differential equations with constant coefficients using the differential operator D, by using this method.

Is it possible to use a similar method (using the differential operator) to solve more advanced ODEs? I'm thinking of both more advanced linear ODEs, such as Euler-Cauchy differential equations, as well as non-linear ODEs.

Are there any articles on the web on this topic, or even textbooks that use this method to solve ODEs?