Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
The term in x ‡ is obtained by writing y ˆ a ‡1 x ‡‡1 in ®rst bracket and
y ˆ a x ‡ in the second. Equating to zero the coecient of the term obtained in
this way we have
giving, with replaced by n,
f4… ‡ ‡ 1†… ‡ †‡2… ‡ ‡ 1†ga ‡1 ‡ a ˆ 0;
1
a n‡1 ˆ
2… ‡ n ‡ 1†…2 ‡ 2n ‡ 1† a n:
This relation is true for n ˆ 1; 2; 3; ... and is called the recurrence relation for the
coecients. Using the ®rst root ˆ 0 of the indicial equation, the recurrence
relation gives
and hence
a n‡1 ˆ
1
2…n ‡ 1†…2n ‡ 1† a n
a 1 ˆ a 0
2 ; a 2 ˆ a 1
12 ˆ a0
4! ; a 3 ˆ a 2
30 ˆa 0
6! ; ...:
Thus one solution of the di€erential equation is the series
!
a 0 1 x 2! ‡ x2
4! x3
6! ‡ :
With the second root ˆ 1=2, the recurrence relation becomes
1
a n‡1 ˆ
…2n ‡ 3†…2n ‡ 2† a n:
Replacing a 0 (which is arbitrary) by b 0 , this gives
a 1 ˆ b 0
3 2 ˆb 0
3! ; a 2 ˆ a 1
5 4 ˆ b0
5! ; a 3 ˆ a 2
7 6 ˆb 0
7! ; ...:
and a second solution is
b 0 x 1=2
1 x 3! ‡ x2
5! x3
7! ‡ !
:
The general solution of the equation is a linear combination of these two solutions.
Many physical problems require solutions which are valid for large values of
the independent variable x. By using the transformation x ˆ 1=t, the di€erential
equation can be transformed into a linear equation in the new variable t and the
solutions required will be those valid for small t.
In Example 2.14 the indicial equation has two distinct roots. But there are two
other possibilities: (a) the indicial equation has a double root; (b) the roots of the
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