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If y=y(x) satisfies the differential equ...

If `y=y(x)` satisfies the differential equation `8sqrt(x)(sqrt(9+sqrt(x)))dy=(sqrt(4+sqrt(9+sqrt(x))))^(-1)dx ,x >0` and `y(0)=sqrt(7,)` then `y(256)=`

A

3

B

9

C

16

D

80

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `dy=(dx)/(8sqrt(x)(sqrt(9+sqrt(x))(sqrt(4+sqrt(9+sqrt(x))))))`
Let `4+sqrt(9+sqrt(x))=t`
`rAr 1(2sqrt(9+sqrt(x))).1/(2sqrt(x))dx=dt`
`rArr dy=(dt)/(2sqrt(t))`
`rArr 2dy=1/sqrt(t)dt`
`rArr 2y=2sqrt(t)+c`
`rArr 2y=2sqrt(4+sqrt(9+sqrt(x)))`
Given, `y(0) = sqrt(7) rArr c=0`
`therefore y=sqrt(4+sqrt(9+sqrt(x)))`
`therefore y(256)=3`
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