# Nonlinear Second-Order ODEs

The general form of a nonlinear second-order ODE is

For simplicity, assume that the equation can be solved for the highest-order derivative to give

There are a few classes of nonlinear second-order ODEs for which solutions can be easily found.

The first class consists of equations that do not explicitly depend on ; that is, equations of the form . Such equations can be regarded as first-order ODEs in .

As in the case of linear second-order ODEs, the solution depends on two arbitrary parameters C[1] and C[2].

The second class of easily solvable nonlinear second-order equations consists of equations that do not depend explicitly on or ; that is, equations of the form . These equations can be reduced to first-order ODEs with independent variable . Inverse functions are needed to give the final solution for .

The third class consists of equations that do not depend explicitly on ; that is, equations of the form . Once again, these equations can be reduced to first-order ODEs with independent variable .

The fourth class consists of equations that are homogeneous in some or all of the variables , , and . There are several possibilities in this case, but here only the following simple example is considered.

The fifth and final class of easily solvable nonlinear second-order ODEs consists of equations that are exact or can be made to be exact using an integrating factor.

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It is important to note that the solutions to fairly simple-looking nonlinear ODEs can be complicated. Verifying and applying the solutions in such cases is a difficult problem.