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Overview of Linear Second-Order ODEs

Solving linear first-order ODEs is straightforward and only requires the use of a suitable integrating factor. In sharp contrast, there are a large number of methods available for handling linear second-order ODEs, but the solution to the general equation belonging to this class is still not available. Therefore the linear case is discussed in detail before moving on to nonlinear second-order ODEs.
The general linear second-order ODE has the form
Here, P (x), Q (x) and R (x) are arbitrary functions of x. The term "linear" refers to the fact that the degree of each term in y (x), y (x) and y (x) is 1. (Thus, terms like y (x)2 or y (x)y (x) would make the equation nonlinear.)