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 order ordinary differential can be similarly reduced to
Common numerical methods for solving initial value problems of ordinary differential equations are summarized:
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.
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.
order ordinary differential can be similarly reduced to
