Appendix: Problem Index
605
(a) Area enclosed by curve y = x sin x and x-axis
(b) Volume generated when curve rotates about x-axis
1.54
1.2.8 Ordinary Differential Equations
dy/dx = (x
3
+ y
3 )/3x y
2
1.55
d
3 y/dx
3
− 3d
2 y/dx
2
+ 4y = 0
1 . 5 6
d
4 y/dx
4
− 4d
3 y/dx
3
+ 10d
2 y/dx
2
− 12dy/dx + 5y = 0
1 . 5 7
d
2 y/dx
2
+ m
2 y = cos ax
1.58
d
2 y/dx
2
− 5dy/dx + 6y = x
1.59
Equation of motion for damped oscillator
1.60
Modes of oscillation of coupled springs
1.61
SHM of a rolling cylinder with a spring attached to it
1.62
d
2 y/dx
2
− 8dy/dx = −16y
1.63
x
2 dy/dx + y(x + 1)x = 9x
2
1.64
d
2 y/dx
2
+ dy/dx − 2y = 2 cosh (2x)
1 . 6 5
xdy/dx − y = x
2
1.66
(a) y
− 2y/x = 1/x
2 (b) y
+ 5y
+ 4y = 0
1 . 6 7
(a) dy/dx + y = e
−x (b) d
2 y/dx
2
+ 4y = 2 cos (2x)
1 . 6 8
dy/dx + 3y/(x + 2) = x + 2 with boundary conditions
1.69
(a) d
2 y/dx
2
− 4dy/dx + 4y = 8x
2
− 4x − 4, with boundary
conditions
(b) d
2 y/dx
2
+ 4y = sin x
1.70
d
3 y/dx
3
− d
2 y/dx
2
+ dy/dx − y = 0
1 . 7 1
1.2.9 Laplace Transforms
Solution of radioactive chain decay
1.72, 73
(a) £(e
ax ) = 1/(s − a) (b) £(cos ax) = s/(s
2
+ a
2 )
(c) £(sin ax) = a/(s
2
+ a
2 )
1.74
1.2.10 Special Functions
For Hermite polynomials (a) H
n = 2n H n−1
(b) H n+1 = 2ξH n − 2n H n−1
1.75
For Bessel function (a)
d
dx
[x
n J n (x)] = x
n J n−1 (x)
(b)
d
dx
[x
−n J n (x)] = −x
−n J n+1 (x)
1.76
(a) J n−1 (x) − J n+1 (x) = 2
d
dx
J n (x)
(b) J n−1 (x) + J n+1 (x) = 2(n/x)J n (x)
1.77
(a) J 1/2 (x) =
2
π x
sin x (b) J −1/2 (x) =
2
π x
cos x
1.78
For Legendre polynomials
1
−1 p n (x) p m (x)dx =
2
2n+1
, if m = n
and = 0 if m = n
1.79
For large n and small θ, p n (cos θ) ≈ J 0 (nθ)
1 . 8 0
(a) (l + 1) p l+1 = (2l + 1)x p l − lp l−1
(b) P l (x) + 2x p l
(x) = p l
+1 (x) + p l
−1 (x)
1 . 8 1
For Laguerre’s polynomials L n (0) = n!
1 . 8 2
1.2.11 Complex Variables
c dZ /(Z − 2) where C is (a) Circle |Z | = 1 (b) Circle |Z + i| = 3 1.83
c (4Z
2
− 3Z + 1)(Z − 1)
−3 dZ , C encloses Z = 1
1 . 8 4