1000 Solved Problems in Modern Physics

64 1 Mathematical Physics

1.67 (a)y′−

2 y

x

=

1

x^3

(1)

Lety=px,y′=p+xp′

Then (1) becomes

xp′−p= 1 /x^3

Nowddx

(p

x

)

=xpx− 2 p

∴xp′−p=x^2

d

dx

(p

x

)

=

1

x^3

d

dx

(p

x

)

=

1

x^5

or d

(p

x

)

=

dx

x^5

Integrating

p

x

=−

1

4 x^4

+C

or

y

x^2

=−

1

4 x^4

+C

y=−

1

4 x^2

+Cx^2

It is inhomogeneous, first order.

(b)y′′+ 5 y′+ 4 y= 0

D^2 + 5 D+ 4 = 0

(D+4)(D+1)= 0

D=− 4 ,− 1

y=Ae−^4 x+Be−x

It is inhomogeneous, second order.

1.68 (a)

dy

dx

+y=e−x

Compare with the standard equation

dy

dx

+py=Q

P=1;Q=e−x

yexp

(∫

pdx

)

=

[∫

Qexp

(∫

pdx

)]

dx+C

yexp

(∫

1dx

)

=

[∫

e−xexp

(∫

1dx

)]

dx+C

yex=x+C

y=xe−x+Ce−x

(b)d

(^2) y
dx^2

• 4 y=2 cos(2x)(1)
  The complimentary function is obtained fromy′′+ 4 y= 0
  y=U=C 1 sin 2x+C 2 cos 2x
  Differentiate (1) twice