Further Examples of Exactly Solvable HigherOrder Equations
The solutions to many secondorder ODEs can be expressed in terms of special functions. Solutions to certain higherorder ODEs can also be expressed using
AiryAi,
BesselJ, and other special functions.
The solution to this thirdorder ODE is given by products of Airy functions.
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The solution to this thirdorder 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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This verifies that the solution is correct using numerical values.
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As for secondorder linear ODEs, there are modern algorithms for solving higherorder ODEs with rational coefficients. These algorithms give "rationalexponential" 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
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.
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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 thirdorder ODE with no explicit dependence on x or y[x]. It is solved by reducing the order to 2 using a simple integration.
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