When numerical methods provide the solution

Ever since differential equations emerged in the 18^th century as a means of solving problems in physics, numerical methods have been used to seek approximate solutions. The first such method, dating from the 1760s, was devised by Leonhard Euler.

A method devised by Euler ----------------------------

Let us consider the differential equation y’ = f (x, y) on the interval [a, b], with the initial condition y (a) = y*₀, where y₀ is given (for example, by the physics of the problem under consideration or by a known boundary condition…). The function f is a function of two variables, while y is, of course, the unknown function, depending on a variable x. In fact, the notation "y’ = f (x, y)" is shorthand for all the equalities y’(x) = f (x, y(x)), where x belongs to [a, b]. The function f must be "sufficiently regular" for the equation to have a unique solution. Continuity is generally not enough.

For example, on the interval I = [0, 100], the differential equation y’ = 3y^2/3, which can be written as $y^{\mathrm{\prime }}=\sqrt[3]{y^2},y\text{'}\; =\; \backslash sqrt[3]\{y^2\},$ with the initial condition y (0) = 0, has two solutions: y(t) = 0 (the zero function) and y (t) = t³ (the cubic function) for t in I. General theorems give sufficient conditions on f to guarantee the existence and uniqueness of the solution.