1. Formulating Cauchy Problems for ODEs

Adifferential equation is an equation in which the unknown is a function (e.g., y ( t )) and which contains a derivative of this function (e.g., the first derivative y ( t )).

Differential equations are divided into two types: ordinary differential equations (ODEs) and partial differential equations. An ordinary differential equation is an equation that only contains derivatives of one variable: . A partial differential equation contains functions of multiple independent variables and their derivatives: , .

Ordinary differential equations are much easier to solve, both numerically and analytically, than partial differential equations. Solving ODEs, however, has its own share of pitfalls, some of which are discussed in this chapter. The problem of finding solutions to ODEs consists of setting the relation to 0, including derivatives (possibly of different orders) of the unknown function y(x):

The most often-encountered equations are of the following type:

This is called the Cauchy, or standard, form.

Thus, the unknown quantity in ODEs is not a number as in algebraic equations, but some function y(x). According to the differential equation theory, for an ODE to have a solution, initial and boundary conditions have to be specified. Initial and boundary conditions are discussed in more detail when specific examples are considered.

As is well known, the problem of solving ODEs containing higher-order derivatives can be reduced to an ODE system with one derivative. For this, derivatives of the y(x)