64
1 Mathematical Physics
1.67 (a) y
−
2y
x
=
1
x 3
(1)
Let y = px, y
= p + x p
Then (1) becomes
x p
− p = 1/x
3
Now
d
dx
p
x
=
xp− p
x 2
∴ x p
− 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
4x 4 + C
or
y
x 2 = −
1
4x 4 + C
y = −
1
4x 2 + C x
2
It is inhomogeneous, first order.
(b) y
+ 5y
+ 4y = 0
D
2
+ 5D + 4 = 0
(D + 4)(D + 1) = 0
D = −4, −1
y = A e
−4x
+ B e
−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
y exp
pdx
=
Q exp
p dx
dx + C
y exp
1 dx
=
e
−x exp
1 dx
dx + C
ye
x
= x + C
y = xe
−x
+ Ce
−x
(b) d
2 y
dx 2 + 4y = 2 cos(2x)
( 1 )
The complimentary function is obtained from y
+ 4y = 0
y = U = C 1 sin 2x + C 2 cos 2x
Differentiate (1) twice
1 Mathematical Physics
1.67 (a) y
−
2y
x
=
1
x 3
(1)
Let y = px, y
= p + x p
Then (1) becomes
x p
− p = 1/x
3
Now
d
dx
p
x
=
xp− p
x 2
∴ x p
− 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
4x 4 + C
or
y
x 2 = −
1
4x 4 + C
y = −
1
4x 2 + C x
2
It is inhomogeneous, first order.
(b) y
+ 5y
+ 4y = 0
D
2
+ 5D + 4 = 0
(D + 4)(D + 1) = 0
D = −4, −1
y = A e
−4x
+ B e
−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
y exp
pdx
=
Q exp
p dx
dx + C
y exp
1 dx
=
e
−x exp
1 dx
dx + C
ye
x
= x + C
y = xe
−x
+ Ce
−x
(b) d
2 y
dx 2 + 4y = 2 cos(2x)
( 1 )
The complimentary function is obtained from y
+ 4y = 0
y = U = C 1 sin 2x + C 2 cos 2x
Differentiate (1) twice