Reduction of a Higher Order Equation to a System of First Order Equations:

One common technique for handling equations of the form (4.1) is to transform them into an equivalent first order system. If we define the variables

(4.4)

we can write (4.1) as

(4.5a)

while by differentiation of (4.4) for we get

(4.5b)

(4.5) is a system of p first order equations which can be handled by the methods already discussed. The original problem (4.1) will have initial values specified for so (4.5) will have initial values specified for as required.