Often, the variable t means time, and the di erential equations expresses the rule or law of nature which governs the change in the system being studied. Many problems in science and technology lead to di erential equations.
However, for most di erential equations, one must be content with approximate solutions.
An important special case is when f is a linear function of y. In mathematics courses, one learns how to determine exact solutions to this problem for certain special functions f. To start with, we shall study the following problem: given a function f(t y), nd a function y(t) which for a t b is an approximate solution to the initial value problem for the ordinary di erential equation or, with an established abbreviation, the ODE, dy = f(t y) y(a) = c: (13.1.1) dt Theoretical Background 13.1.1 Introduction Ordinary Di erential Equations 13.1 Initial Value Problems for ODEs.
