Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
2.15 Solve the equation 4x 2 y 00 ‡ 4xy 0 ‡…x 2 1†y ˆ 0 by using Frobenius'
method, where y 0 ˆ dy=dx, andy 00 ˆ d 2 y=dx 2 .
2.16 Find a series solution, valid for large values of x, of the equation
…1 x 2 †y 00 2xy 0 ‡ 2y ˆ 0:
2.17 Show that a series solution of Airy's equation y 00 xy ˆ 0is
!
y ˆ a 0 1 ‡ x3
2 3 ‡ x 6
2 3 5 6 ‡
‡ b 0 x ‡ x4
3 4 ‡ x 7
3 4 6 7 ‡ !
:
2.18 Show that Weber's equation y 00 ‡…n ‡ 1 2 1 4 x2 †y ˆ 0 is reduced by the substitution
y ˆ e x2 =4 v to the equation d 2 v=dx 2 x…dv=dx†‡nv ˆ 0. Show
that two solutions of this latter equation are
v 2 ˆ x
v 1 ˆ 1 n 2! x2 ‡
…n 1†
x 3 ‡
3!
n…n 2†
x 4
4!
…n 1†…n 3†
x 5
5!
2.19 Solve the following simultaneous equations
)
Dx ‡ y ˆ t 3
…D ˆ d=dt†:
Dy x ˆ t
2.20 Evaluate the integrals:
(a)
(b)
(c)
(d)
Z 1
0
Z 1
0
Z 1
0
Z 1
0
x 3 e x dx:
x 6 e 2x dx
p
y e
y 2 dy
dx
p
ln x
…hint: let y ˆ 2x†:
…hint: let y 2 ˆ x†:
…hint : let ln x ˆ u†:
2.21 (a) Prove that B…p; q† ˆ2
(b) Evaluate the integral
Z =2
0
Z 1
0
n…n 2†…n 4†
x 6 ‡;
6!
…n 1†…n 3†…n 5†
x 7 ‡:
7!
sin 2p1 cos 2q1 d.
x 4 …1 x† 3 dx:
p
2.22 Show that n!
2n n n e n . This is known as Stirling's factorial approximation
or asymptotic formula for n!.
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