Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.6 DIFFERENTIAL EQUATIONS

but they may be summarised as (i) insufficiently precise approximations to the

derivatives and (ii) inherent instability due to rounding errors.

27.6.2 Taylor series solutions

Since a Taylor series expansion is exact if all its terms are included, and the limits

of convergence are not exceeded, we may seek to use one to evaluatey 1 ,y 2 ,etc.

for an equation

dy

dx

=f(x, y), (27.66)

when the initial valuey(x 0 )=y 0 is given.

The Taylor series is

y(x+h)=y(x)+hy′(x)+

h^2

2!

y′′(x)+

h^3

3!

y(3)(x)+···. (27.67)

In the present notation, at the pointx=xithis is written

yi+1=yi+hy(1)i +

h^2

2!

y(2)i +

h^3

3!

yi(3)+···. (27.68)

But, for the required solutiony(x), we know that

yi(1)≡

(

dy

dx

)

xi

=f(xi,yi), (27.69)

and the value of the second derivative atx=xi,y=yican be obtained from it:

y(2)i =

∂f

∂x

+

∂f

∂y

dy

dx

=

∂f

∂x

+f

∂f

∂y

. (27.70)

This process can be continued for the third and higher derivatives, all of which

are to be evaluated at (xi,yi).

Having obtained expressions for the derivativesy(in)in (27.67), two alternative

ways of proceeding are open to us:

(i) equation (27.68) is used to evaluateyi+1, the whole process is repeated to

obtainyi+2,andsoon;

(ii) equation (27.68) is applied several times but using a different value ofh

each time, and so the corresponding values ofy(x+h) are obtained.

It is clear that, on the one hand, approach (i) does not require so many terms of

(27.67) to be kept, but, on the other hand, theyi(n) have to be recalculated at

each step. With approach (ii), fairly accurate results forymay be obtained for

values ofxclose to the given starting value, but for large values ofha large

number of terms of (27.67) must be kept. As an example of approach (ii) we

solve the following problem.