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 firstorder equations can be expressed by:
and the higherorder equations can be written:
y^{(n)} = f (x, y, y', y'',..., y^{(n1)}) 
The task is to determine the necessary boundary conditions and the relationships between x and y.
A common way of handling a second or higherorder equations is to replace it with an equivalent system
of firstorder equations. The higherorder equation above can always be transformed into a set of n first
order equations. Using the notation:
y_{0} = y
y_{1} = y'
y_{2} = y''
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·
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y_{(n1)} = y^{(n1)} 
then the equivalent firstorder equations become:
y'_{0} = y_{1}
y'_{1} = y_{2}
y'_{2} = y_{3}
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·
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y'_{n} = f ( x, y_{0}, y_{1},..., y_{n1}) 
