Solving ordinary differential equations using the differential operator D

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?

• Sure; for example, the Laplace equation in two variables can be solved by factoring d/dx^2 + d/dy^2 as (d/dx + i d/dy)(d /dx - i d/dy), which gives a direct connection to the Cauchy-Riemann equations. Commented Feb 23, 2011 at 20:46

Solving ordinary differential equations by differential operators can be essentially important. The most fundamental theory I assume can be found here Differential Operator. Note that Heaviside's pioneering work as mentioned by Bill Dubuque needs some patience or caveat to fully master otherwise can easily lead to pitfalls. The power of linearity of ODE can be utilized by the differential operator (also a linear operator) which naturally implies the additivity ($$f(x+y)=f(x)+f(y)$$) and homogeneity ($$f(\alpha x) = \alpha f(x),~\forall\alpha$$).