Classification of Differential Equations
While differential equations have three basic types—ordinary (ODEs), partial (PDEs), or differential-algebraic (DAEs), they can be further described by attributes such as order, linearity, and degree. The solution method used by DSolve and the nature of the solutions depend heavily on the class of equation being solved.
The previous equation is also homogeneous: all terms contain or derivatives of and its right-hand side is zero. Adding a function of the independent variable makes the equation inhomogeneous. The general solution to an inhomogeneous equation with constant coefficients is obtained by adding a particular integral to the solution to the corresponding homogeneous equation.
When the coefficients of an ODE depend on , the ODE is said to have variable coefficients. Since equations with variable coefficients that are rational functions of have singularities that are easily classified, there are sophisticated algorithms available for solving them.
There is a close relationship between functions and differential equations. Starting with a function of almost any type, it is possible to construct a differential equation satisfied by that function. Conversely, any differential equation gives rise to one or more functions, in the form of solutions to that equation. In fact, many special functions from classical analysis have their origins in the solution of differential equations. Mathieu functions are one such class of special functions. Mathieu was interested in studying the vibrations of elliptical membranes. The eigenfunctions for the wave equation that describes this motion are given by products of Mathieu functions.
The examples in this tutorial have focused on the classification of ODEs. The classification of PDEs is similar but more involved. PDEs can also be classified by linearity or nonlinearity, order, degree, and constant or variable coefficients. More important is the classification that identifies a PDE as hyperbolic, parabolic, or elliptic. These classifications are discussed in further detail in "Second-Order PDEs".