General Approach

Most of the methods seen above can be applied to system of ODE's, or, indeed, higher order ODE's. The following approach can easily be extended to an ODE of any degree or to a system of more than 2 ODE's.

If the initial value problem is given by an ODE of higher order, one can rewrite it as a system of first order ODE's. For example, the initial value problem

,

can be written as a system of linear ODE's by the introduction of the variable :

,

,

.

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The coefficients may be functions of ! This system can be rewritten in the us more familiar form

,

,

.

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We can generalise the previous methods to solve a system of ODE's of the form above. For example, the family of Taylor series methods and Runge-Kutta methods can be generalised as

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,

,

where the are approximations for the values and respectively.

In case of the Runge-Kutta method of order four, this translates as:

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,

,

,

,

,

,

,

,

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