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Unformatted text preview: Mathematics 421 Solutions 13.6 Spring 2008 Overview The use of separation of variables to identify solutions of a partial differential equa- tion that are products of functions of the individual variables relies on the equation being linear and homogeneous to construct general solutions from these pieces. If the equation of not homogeneous, one aims to find a particular solution so that all solutions differ from this one by a solution of a related ho- mogeneous equation. Often, this particular solution is a product of functions of the individual variables, but there are many more techniques for finding a solution than are described in the examples given in the textbook. More examples are needed, since one cannot claim to have a method if one needs some clever insight to get started. We shall assume that we are interested in solving an equation in which the independent variables are x and t where one is given boundary conditions for x and initial conditions for t . An important first step is to assure that the boundary conditions are met. Once this is done, the solution will have an eigenfunction expansion in x with coefficients depending on t . These coefficients can be expressed as the solution of an inhomogeneous linear ordinary differential equation with initial conditions. The solution of the corresponding homogeneous equation will involve the functions obtained by the separation of variables method, and there will also be a particular solution . Methods for finding the particular solution are: (1) guessing the form of the solution and using undetermined coefficients ; (2) modifying the homogeneous solution using variation of parameters ; or (3) finding the solution by Laplace transforms . The text emphasizes method (1). Usually, method (2) is complicated, so it is never attempted for the equations with constant coefficients that we meet in these applications. Laplace transforms often provided a superior organization of initial value problems , so the failure to suggest them is an oversight. A general method to solve inhomogeneous linear partial differential equations with a boundary con- dition on one variable is to identify the eigenfunctions of the one dimensional boundary value problem associated with that variable and to express the solution as an eigenfunction expansion whose coefficients depend on the other variable (our examples here all have only two independent variables, but similar meth- ods apply to equations with more variables). This approach agrees in spirit with the method of variation of parameters....
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This note was uploaded on 09/29/2009 for the course 650 421 taught by Professor Bumby during the Spring '08 term at Rutgers.
- Spring '08