Differential Equations

Abstract

Maple can solve many ordinary differential equations analytically in explicit and implicit form. Traditional techniques such as the method of Laplace transformations, integrating factors, etc., as well as more advanced techniques like the method of Lie symmetries are available through the ordinary differential equation solver dsolve. Lie symmetries enable to generate new solutions from a particular solution of a differential equation; they form a systematic way of solving differential equations, in particular nonlinear differential equations. The procedure pdsolve provides classical methods to solve partial differential equations such as the method of characteristics and techniques for uncoupling systems of equations. Lie symmetry methods are also available for partial differential equations, viz., in the liesymm package. Approximate methods such as Taylor series and power series methods have been implemented for ordinary differential equations. And if all fails, one can still use the numerical solver based on the Runge-Kutta method or other numerical methods. The DEtools package contains procedures for graphical presentation of solutions of differential equations, and utilities such as change of variables (dependent as well as independent variables) or computation of Lie symmetries of ordinary differential equations. The PDEtools package play a similar role for partial differential equations. Moreover, Maple provides all the tools to apply perturbation methods, like the Poincaré-Lindstedt method and the method of multiple scales up to high order. In this chapter, we shall discuss the tools available in Maple for studying differential equations. Many examples come from applied mathematics.