Approximate Methods for First Order Differential Equations

In a Nut Shell: Not all first order differential equations (linear or nonlinear)

dy/dx = f(x,y)

can be solved exactly and explicitly. For example you may not be able to integrate f(x,y)

to obtain y(x). In such cases there are several approximate methods that you can use.

A number of these are:

Picard's Method	Click here for a discussion of Picard's Method. Click here for examples.
Euler's Method	Click here for a discussion of Euler's Method. Click here for an example.
The Runge-Kutta Method	Click here for a discussion of the Runge-Kutta Method. Click here for an example.

Problem Statement: Find an approximate solution to the first order, ordinary d.e.

dy/dx = f [ (x,y(x) ] subject to the condition y(a) = b

Note: Approximate methods apply to first order differential equations that

may be linear or nonlinear.

Note: Each numerical method may contain local and cumulative errors.

Note: You need to decide on an appropriate step size for each method.

Larger step sizes get you to the numerical solution quicker but may

not be as accurate.

Smaller step sizes will take longer but each iteration carries with it

a local error which may accumulate.

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