Proof of Formulae Used to Solve Differential Equations

The Taylor series for a function expanded about a pointis

We can derive two very useful series, evaluated atand

(1)

(2)

(1)-(2) gives (3)

(1)+(2) gives(4)

(3) and (4) may be used to solve second order differential equations numerically at any point, given any two boundary conditionsandWrite

and

Then (3) becomesand (4) becomes

We can substitute this, withinto the second order differential equationto obtain

This equation is rearranged to makethe subject:

Givenandand a functionwe can find