# 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.

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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 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.

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.