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

Options

A. 3
B. 9
C. 16
D. 80

Answer

A

Explanation

The givne differential equaiton is
`8sqrtx(sqrt(9+sqrtx))dy=(sqrt(4+sqrt(9+sqrtx)))^(-1)dx`
`rArr" "dy=(1)/({8-sqrtx(sqrt(9+sqrtsqrtx))}{sqrt(4+sqrt(9+sqrtx))})dx`
Let `4+sqrt(9+sqrtx))=t`, Then `(1)/(2sqrt(9+sqrtx))xx(1)/(2sqrtx)xx(1)/(2sqrtx)dx=dt`
Substituting `4+sqrt(9+sqrtx))=t` in the given differential equation,
we obtain `dy = (1)/(2sqrtt)d t`. Integrating, we obtain
`y=sqrtt+CrArry=sqrt(4+sqrt(9+sqrtx))=C" ...(i)"`
It is given that `y=sqrt7` when x = 0. Substituting these values in (i), we get C = 0.