
5-16 Calculus – Differentiation and Integration
Example 5.22 If y = A cos(log x) + B sin(log x), then show that x
2
y
n+2
+ (2n + 1)xy
n+1
+ (n
2
+ 1)y
n
= 0.
Solution: Given y A x B x= +cos(log ) sin(log ).
Now,
y A x
x
B x
x
xy A
1
1
1 1
= −
+
= −
sin(log ) cos(log )
sin((log ) cos(log ).x B x+
On differentiating the above equation, we obtain
xy y A x
x
B x
x
x y
2 1
2
2
1 1
+ = −
−
⇒
cos(log ) sin(log )
++ = − +
⇒ + = −
⇒ + + =
xy A x B x
x y xy y
x y xy y
1
2
2 1
2
2 1
0
[ cos(log ) sin(log )]
.
Differentiating the above equation for n times using Leibnitz’s theorem, we get
D y x D y x D y
D y x nD y x
n n
D
n n n
n n n
( ) (