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Common Numerical Methods for Solving ODE's

The numerical methods for solving ordinary differential equations are methods of integrating a system of first order differential equations, since higher order ordinary differential equations can be reduced to a set of first order ODE's. For example,

An nth order ordinary differential can be similarly reduced to

Common numerical methods for solving initial value problems of ordinary differential equations are summarized:

• Euler Method

• Midpoint Method

• Runge-Kutta Method

Euler Method

The Euler method is important in concept for it points the way of solving ODE by marching a small step at a time on the right-hand-side to approximate the "derivative" on the left-hand-side.

However, the Euler method has limited value in practical usage.

Midpoint Method

The midpoint method, also known as the second-order Runga-Kutta method, improves the Euler method by adding a midpoint in the step which increases the accuracy by one order.

Runge-Kutta Method

The fourth-order Runge-Kutta method is by far the ODE solving method most often used. It can be summarized as follows: