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
, and other special functions.
The solution to this third-order ODE is given by products of Airy functions.
The solution to this third-order ODE is given by Bessel functions.
This plot shows the oscillatory behavior of the solutions on different parts of the real line.
The solution to this fourth-order linear ODE is expressed in terms of HypergeometricPFQ
This verifies that the solution is correct using numerical values.
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.
The equations considered so far have been homogeneous; that is, with no term free of y (x)
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.
This is the general solution to the homogeneous equation.
This particular solution is part of the general solution to the inhomogeneous equation.
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 x
. It is solved by reducing the order to 2 using a simple integration.