# Further Examples of Exactly Solvable Higher-Order Equations

The solutions to many second-order ODEs can be expressed in terms of special functions. Solutions to certain higher-order ODEs can also be expressed using AiryAi, BesselJ, and other special functions.

The solution to this third-order ODE is given by products of Airy functions.

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The solution to this third-order ODE is given by Bessel functions.

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This plot shows the oscillatory behavior of the solutions on different parts of the real line.

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The solution to this fourth-order linear ODE is expressed in terms of

HypergeometricPFQ.

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This verifies that the solution is correct using numerical values.

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As for second-order linear ODEs, there are modern algorithms for solving higher-order ODEs with rational coefficients. These algorithms give "rational-exponential" solutions, which are combinations of rational functions and exponentials of the integrals of rational functions. These algorithms are combined with techniques such as reduction of order to produce a complete solution for the given ODE.

The general solution to this equation has a rational term and terms that depend on Airy functions. The Airy functions come from reducing the order of the equation to 2.

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The equations considered so far have been homogeneous; that is, with no term free of or its derivatives. If the given ODE is inhomogeneous, DSolve applies the method of *variation of parameters* to obtain the solution.

Here is an example of this type. The exponential terms in the solution come from the general solution to the homogeneous equation, and the remaining term is a particular solution (or particular integral) to the problem.

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This is the general solution to the homogeneous equation.

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This particular solution is part of the general solution to the inhomogeneous equation.

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Thus, the general solution for the inhomogeneous equation is the sum of the general solution to the homogeneous equation and a particular integral of the ODE.

The solution methods for nonlinear ODEs of higher order rely to a great extent on reducing the problem to one of lower order.

Here is a nonlinear third-order ODE with no explicit dependence on

or

. It is solved by reducing the order to 2 using a simple integration.

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