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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?

haroba
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    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. – Qiaochu Yuan Feb 23 '11 at 20:46

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There is a large literature on "operational calculus" dating back to Heaviside's pioneering work. One particularly powerful operator factorization technique is the Infeld - Hull ladder method - which plays a big role in unifying many classes of special functions that arise in physics (mainly via separation of variables in various coordinate systems). Willard Miller showed that this method is equivalent to the representation theory of four local Lie groups. This lie-theoretic approach served to powerfully unify and "explain" all prior similar attempts to provide a unfied theory of such classes of special functions. See my post here for further details and references.

Community
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Bill Dubuque
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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 ($f(\alpha x) = \alpha f(x),~\forall\alpha$).

In addition, all the properties of differentiation rules in calculus will apply to the operators, but there are other peculiarities as well. For instance Fourier and Lalpace transform operators can also be good references.

MathArt
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I am also an enthusiast of this technique solve a differential equation (See this link). With the operator D, we find the general solution of non-linear equations homogenêneas with f being a polynomial, trigonometric functions, exponential, or combinations of these. You can also define a differential operator to study the differential equations of Euler-Cauchy type.

Mathsource
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