Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
and, with ˆ p ‡ q, a ˆ 1 ‡ v, this can be written
…p ‡ q†…1 ‡ v† pq ˆ
Z 1
0
e …1‡v†t t p‡q1 dt:
Multiplying by v p1 and integrating with respect to v between 0 and 1,
…p ‡ q†
Z 1
v p1 …1 ‡ v† pq dv ˆ
Z 1
v p1 dv
Z 1
0
0
0
e …1‡v†t t p‡q‡1 dt:
Then interchanging the order of integration in the double integral on the right and
using Eq. (2.42),
…p ‡ q†B…p; q† ˆ
ˆ
Z 1
e t t p‡q1 dt
Z 1
0
0
Z 1
0
ˆ …p†
e vt v p1 dv
e t t p‡q1 …p†
t p dt; using Eq: …2:44†
Z 1
0
e t t q1 dt ˆ …p†…q†:
Example 2.15
Evaluate the integral
Z 1
0
3 4x2 dx:
Solution:
We ®rst notice that 3 ˆ e ln 3 , so we can rewrite the integral as
Z 1
0
3 4x2 dx ˆ
Z 1
0
…e ln 3 † …4x2† dx ˆ
Z 1
0
e …4ln3†x2 dx:
Now let (4 ln 3)x 2 ˆ z, then the integral becomes
Z !
1
e z z 1=2
Z
1 1
d p
ˆ p
4ln3 2 z 1=2 e z dz ˆ …1 2
p
†
4ln3
2 ˆ
4ln3
0
0
p
p
2 :
4ln3
Problems
2.1 Solve the following equations:
(a) xdy=dx ‡ y 2 ˆ 1;
(b) dy=dx ˆ…x ‡ y† 2 .
2.2 Melting of a sphere of ice: Assume that a sphere of ice melts at a rate
proportional to its surface area. Find an expression for the volume at any
time t.
2.3 Show that …3x 2 ‡ y cos x†dx ‡…sin x 4y 3 †dy ˆ 0 is an exact di€erential
equation and ®nd its general solution.
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