1000 Solved Problems in Modern Physics

Appendix: Problem Index 605

(a) Area enclosed by curvey=xsinxandx-axis

(b) Volume generated when curve rotates aboutx-axis 1.54

1.2.8 Ordinary Differential Equations

dy/dx=(x^3 +y^3 )/ 3 xy^2 1.55

d^3 y/dx^3 −3d^2 y/dx^2 + 4 y=01.56

d^4 y/dx^4 −4d^3 y/dx^3 +10d^2 y/dx^2 −12dy/dx+ 5 y=01.57

d^2 y/dx^2 +m^2 y=cos ax 1.58

d^2 y/dx^2 −5dy/dx+ 6 y=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=− 16 y 1.63

x^2 dy/dx+y(x+1)x= 9 x^2 1.64

d^2 y/dx^2 +dy/dx−^2 y=2 cosh (2x)1.65

xdy/dx−y=x^2 1.66

(a)y′− 2 y/x= 1 /x^2 (b)y′′+ 5 y′+ 4 y=01.67

(a) dy/dx+y=e−x(b) d^2 y/dx^2 + 4 y=2 cos (2x)1.68

dy/dx+ 3 y/(x+2)=x+2 with boundary conditions 1.69

(a) d^2 y/dx^2 −4dy/dx+ 4 y= 8 x^2 − 4 x−4, with boundary

conditions

(b) d^2 y/dx^2 + 4 y=sinx 1.70

d^3 y/dx^3 −d^2 y/dx^2 +dy/dx−y=01.71

1.2.9 Laplace Transforms

Solution of radioactive chain decay 1.72, 73

(a)£(eax)= 1 /(s−a)(b)£(cosax)=s/(s^2 +a^2 )

(c)£(sinax)=a/(s^2 +a^2 )

1.74

1.2.10 Special Functions

For Hermite polynomials (a)Hn′= 2 nHn− 1

(b)Hn+ 1 = 2 ξHn− 2 nHn− 1

1.75

For Bessel function (a)ddx[xnJn(x)]=xnJn− 1 (x)

(b)ddx[x−nJn(x)]=−x−nJn+ 1 (x)

1.76

(a)Jn− 1 (x)−Jn+ 1 (x)= (^2) ddxJn(x)
(b)Jn− 1 (x)+Jn+ 1 (x)=2(n/x)Jn(x)

1.77

(a)J 1 / 2 (x)=

√

2

πxsinx(b)J−^1 /^2 (x)=

√

2

πxcosx 1.78

For Legendre polynomials

∫ 1

− 1 pn(x)pm(x)dx=

2

2 n+ 1 ,ifm=n

and=0ifm

=n

1.79

For largenand smallθ,pn(cosθ)≈J 0 (nθ)1.80

(a) (l+1)pl+ 1 =(2l+1)xpl−lpl− 1

(b)Pl(x)+ 2 xpl′(x)=pl′+ 1 (x)+pl′− 1 (x)1.81

For Laguerre’s polynomialsLn(0)=n!1.82

1.2.11 Complex Variables∮

∮cdZ/(Z−2) whereCis (a) Circle|Z|=1(b)Circle|Z+i|=31.83

c(4Z

(^2) − 3 Z+1)(Z−1)− (^3) dZ,CenclosesZ=11.84