Introduction

The previous sections have been primarily concerned with finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.

The solution and/or its derivatives are required to have specific values at a single point, for example, and . Such problems are traditionally called initial value problems (IVPs) because the system is assumed to start evolving from the fixed initial point (in this case, 0).

The solution is required to have specific values at a pair of points, for example, and . These problems are known as boundary value problems (BVPs) since the points 0 and 1 are regarded as boundary points (or edges) of the domain of interest in the application.

The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations. Since there is no essential distinction between the methods for IVPs and for BPS, both types of problems will be referred to here as boundary value problems.

BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear BVPs. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, the BVP naturally breaks up into simpler BVPs over the regions in which the coefficients are continuous.

The next few sections contain examples to illustrate the solution of boundary value problems of all three types: linear, nonlinear, and piecewise.