MATHEMATICA TUTORIAL

Linear Second-Order ODEs with Rational Coefficients

The hypergeometric functions play a unifying role in mathematical analysis since many important functions, such as the Bessel functions and Legendre functions, are special cases of them. Each hypergeometric function is associated with a linear ODE having rational coefficients.

Here is the ODE for the Hypergeometric2F1 function.
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DSolve can solve a large class of second-order linear ODEs by reducing them to the ODEs for hypergeometric functions. The reduction involves coordinate transformations of both the independent and dependent variables.

This equation is equivalent to the ODE for Hypergeometric2F1.
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This verifies the solution using numerical values.
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Solutions to this equation are returned in terms of HypergeometricU (the confluent hypergeometric function) and LaguerreL. This example appears on (equation 2.16, page 403 of [K59]).
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The ODEs for special functions have been studied since the eighteenth century. During the last 30 years, powerful algorithms have been developed for systematically solving ODEs with rational coefficients. An important algorithm of this type is Kovacic's algorithm, a decision procedure that either generates a solution for the given ODE in terms of Liouvillian functions or proves that the given ODE does not have a Liouvillian solution.

This equation is solved using Kovacic's algorithm.
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The solution returned from Kovacic's algorithm may occasionally include functions such as ExpIntegralEi or an unevaluated integral of elementary functions because, while it is easy to find a second solution for a second-order linear ODE once one solution is known, the integral involved in finding the second solution may be hard to evaluate explicitly.

The solution to this equation is obtained using Kovacic's algorithm. It includes ExpIntegralEi.
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In general, the solutions for linear ODEs with rational coefficients and order greater than one can be given in terms of DifferentialRoot objects. This is similar to the representation for solutions of polynomial equations in terms of Root.

The solution to this equation is given in terms of DifferentialRoot.
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The solution may be evaluated and plotted in the usual way.
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