The Taylor series for a function expanded about a point
is
We can derive two very useful series, evaluated at
and
(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 conditions
and
Write
and
Then (3) becomes
and (4) becomes
We can substitute this, with
into the second order differential equation
to obtain
This equation is rearranged to make
the subject:
Givenand
and a function
we can find