Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
of simultaneous equations may be found by solving for each dependent variable
separately, as shown by the following example
)
Dx ‡ 2y ‡ 3x ˆ 0
…D ˆ d=dt†
3x ‡ Dy 2y ˆ 0
which can be rewritten as
)
…D ‡ 3†x ‡ 2y ˆ 0;
3x ‡…D 2†y ˆ 0:
We then operate on the ®rst equation with (D 2) and multiply the second by a
factor 2:
)
…D 2†…D ‡ 3†x ‡ 2…D 2†y ˆ 0;
6x ‡ 2…D 2†y ˆ 0:
Subtracting the ®rst from the second leads to
…D 2 ‡ D 6†x 6x ˆ…D 2 ‡ D 12†x ˆ 0;
which can easily be solved and its solution is of the form
x…t† ˆAe 3t ‡ Be 4t :
Now inserting x…t† back into the original equation to ®nd y gives:
y…t† ˆ3Ae 3t ‡ 1 2 Be4t :
The gamma and beta functions
The factorial notation n! ˆ n…n 1†…n 2†3 2 1 has proved useful in
writing down the coecients in some of the series solutions of the di€erential
equations. However, this notation is meaningless when n is not a positive integer.
A useful extension is provided by the gamma (or Euler) function, which is de®ned
by the integral
…† ˆ
and it follows immediately that
Integration by parts gives
… ‡ 1† ˆ
Z 1
0
…1† ˆ
Z 1
0
Z 1
0
e x x 1 dx … >0† …2:38†
e x dx ˆ‰e x Š 1 0
ˆ 1: …2:39†
Z 1
e x x dx ˆ‰e x x Š 1 0 ‡ e x x 1 dx ˆ …†:
94
0
…2:40†