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Ordinary Differential Equations

Several numerical methods can be used to solve ordinary differential equations. These equations are especially useful when differential equations cannot be solved analytically.

The general form of the first-order equations can be expressed by:

y' = f (x,y)

and the higher-order equations can be written:

y(n) = f (x, y, y', y'',..., y(n-1))

The task is to determine the necessary boundary conditions and the relationships between x and y.

A common way of handling a second- or higher-order equations is to replace it with an equivalent system of first-order equations. The higher-order equation above can always be transformed into a set of n first order equations. Using the notation:

y0 = y
y1 = y'
y2 = y''
y(n-1) = y(n-1)

then the equivalent first-order equations become:

y'0 = y1
y'1 = y2
y'2 = y3
y'n = f ( x, y0, y1,..., yn-1)

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