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# 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 y (x) 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 y (x); that is, equations of the form y (x)=f (x, y (x)). Such equations can be regarded as first-order ODEs in u (x)=y (x).
Here is an example of this type.
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As in the case of linear second-order ODEs, the solution depends on two arbitrary parameters C[1] and C[2].
Here is a plot of the solution for a specific choice of parameters.
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This verifies the solution.
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The second class of easily solvable nonlinear second-order equations consists of equations that do not depend explicitly on x or y (x); that is, equations of the form y (x)f (y (x)). These equations can be reduced to first-order ODEs with independent variable y. Inverse functions are needed to give the final solution for y (x).
Here is an example of this type.
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The third class consists of equations that do not depend explicitly on x; that is, equations of the form y (x)=f (y (x), y (x)). Once again, these equations can be reduced to first-order ODEs with independent variable y.
This example is based on (equation 6.40, page 550 of [K59]). The underlying first-order ODE is an Abel equation. The hyperbolic functions in the solution result from the automatic simplification of Bessel functions.
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The fourth class consists of equations that are homogeneous in some or all of the variables x, y (x), and y (x). There are several possibilities in this case, but here only the following simple example is considered.
In this equation, each term has a total degree of 2 in the variables y[x], y[x], and y[x]. This equation can be solved by transforming it to a first-order ODE.
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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.
Here is an example of this type, based on (equation 6.51, page 554 of [K59]).
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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.