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 For analytical solutions of ODE, click here. 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 order ordinary differential can be similarly reduced to Common numerical methods for solving initial value problems of ordinary differential equations are summarized:
 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: Glossary 