Many problems in applied mathematics lead to ordinary differential equations. In the simplest case one seeks a differentiable function y = y(x) of one real variable x, whose derivative y′(x) is to satisfy an equation of the form y′(x) = f(x, y(x)), or more briefly, $$y' = f\left( {x,y} \right)$$ (7.0.1) one then speaks of an ordinary differential equation. In general there are infinitely many different functions y which are solutions of (7.0.1). Through additional requirements one can single out certain solutions from the set of all solutions. Thus, in an initial-value problem, one seeks a solution y of (7.0.1) which for given x 0, y 0 satisfies an initial condition of the form $$y\left( {{x_0}} \right) = {y_0}$$ (7.0.2)