Graphic Solution of a Second-Order Differential Equation

This Demonstration shows the Euler–Cauchy method for approximating the solution of an initial value problem with a second-order differential equation. An example of such an equation is , with derivatives from now on always taken with respect to . This equation can be written as a pair of first-order equations, , .

More generally, the method to be described works for any system of two first-order differential equations , with initial conditions , . The particular kinds of systems used as examples here, , reduce to that general type by introducing to get the system , .

The method consists of simultaneously calculating approximations of (cyan) and (green):

,

, .

The pairs are the coordinates of points ,, …, that form the so-called Euler's polygonal line that approximates the graph of the function . In the same way, the pairs are the coordinates of points , , …, that form Euler's polygonal line, which approximates the graph of the function .

The Euler method is the most basic approximation method. The Demonstration compares it with more advanced methods given by the built-in Mathematica function NDSolve.