Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS IN POWER SERIES
Solutions in power series
In many problems in physics and engineering, the di€erential equations are of
such a form that it is not possible to express the solution in terms of elementary
functions such as exponential, sine, cosine, etc.; but solutions can be obtained as
convergent in®nite series. What is the basis of this method? To see it, let us
consider the following simple second-order linear di€erential equation
d 2 y
dx 2 ‡ y ˆ 0:
Now assuming the solution is given by y ˆ a 0 ‡ a 1 x ‡ a 2 x 2 ‡, we further
assume the series is convergent and di€erentiable term by term for suciently
small x. Then
and
dy=dx ˆ a 1 ‡ 2a 2 x ‡ 3a 3 x 2 ‡
d 2 y=dx 2 ˆ 2a 2 ‡ 2 3a 3 x ‡ 3 4a 4 x 2 ‡:
Substituting the series for y and d 2 y=dx 2 in the given di€erential equation and
collecting like powers of x yields the identity
…2a 2 ‡ a 0 †‡…2 3a 3 ‡ a 1 †x ‡…3 4a 4 ‡ a 2 †x 2 ‡ˆ0:
Since if a power series is identically zero all of its coecients are zero, equating to
zero the term independent of x and coecients of x, x 2 ; ...; gives
and it follows that
The required solution is
y ˆ a 0
2a 2 ‡ a 0 ˆ 0; 4 5a 5 ‡ a 3 ˆ 0;
2 3a 3 ‡ a 1 ˆ 0; 5 6a 6 ‡ a 4 ˆ 0;
3 4a 4 ‡ a 2 ˆ 0;
a 2 ˆ a 0
2 ; a 3 ˆ a 1
2 3 ˆa 1
3! ; a 4 ˆ a 2
3 4 a 0
4!
a 5 ˆ a 3
4 5 ˆ a1
5! ; a 6 ˆ a 4
5 6 ˆa 0
6! ; ...:
1 x2
2! ‡ x4
4! x6
6! ‡ !
‡ a 1 x x3
3! ‡ x5
5! ‡ !
;
you should recognize this as equivalent to the usual solution
y ˆ a 0 cos x ‡ a 1 sin x, a 0 and a 1 being arbitrary constants.
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