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Solving Higher Order ODEs using Mathematica 5.2
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| Organization: | Wolfram Research, Inc. |
| Department: | Kernel Technology |
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2005-07-25
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The Mathematica function DSolve has been equipped with several modern algorithms for solving higher order linear ordinary differential equations (ODEs) in Version 5.2. The aim of this notebook is to explain the motivation for these developments and to provide some information and examples which illustrate the new functionality.
In Mathematica 5.1, we had focussed on adding methods for solving first order and second order ODEs such as Abel equations, hypergeometric-type equations and equations with non-rational coefficients using DSolve. As explained in the Advanced Documentation for DSolve, the code structure for this function is hierarchical, so that the problem of solving ODEs of order greater than 2 is often reduced to that of solving a first order or second order ODE. Within the last few years, a deeper understanding of several aspects of higher order ODEs (such as factorization techniques) has emerged which makes it possible to carrry out this reduction in a systematic way. Also, higher order ODEs (particularly orders 3 and 4) are increasingly being seen in scientific models. Thus, we were interested in widening the application of the methods implemented in Version 5.1 to higher order ODEs.
In Section 2, we will review the methods for solving higher order ODEs which were already available in V 5.1.
In Section 3, we will discuss the implementation of the Bronstein-Mulders-Weil-van Hoeij algorithm for solving linear ODES of arbitrary order that are symmetric powers of second order ODEs.
Section 4 deals with a generalization of the notion of symmetric power in which we start with a pair of second order ODEs.
In Section 5, we deal with the important notion of factorization for a differential operator.
We end this introduction by noting that it will be convenient to switch back and forth between (homogeneous) differential equations and the corresponding differential operators since the algorithms really refer to the differential operators and the final step of integrating lower order ODEs to find the solutions is straightforward in all cases.
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Differential equations, ODE
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| SolvingHigherOrderODEsMathematica52April272005.nb (164.1 KB) - Mathematica Notebook [for Mathematica 5.2] |
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