The solution of differential equation
`y dx+ ( 2 sqrt(xy)-x )dy = 0 ` is

Options

A. ` log |y|=sqrt(x/y)=C`
B. ` log|y | +sqrt(x/y)=C`
C. ` log |y| +2sqrt(x/y)=C`
D. None of these

Answer

b

Explanation

Given , ` y dx + ( 2 sqrt(xy)-x) dy = 0 `
` rArr (dy)/(dx) = y/(x-2sqrt(xy))" "` …(i)
This is a homogeneous equation
To simplify it , put y = vx
On differentiating both sides w.r.t x, we get
` :.` From Eq. (i)
` v + x (dv)/(dx) - v/(1-2sqrt(2)) rArr x (dv)/(dx) = (2v^(3//2))/(1-2sqrt(v))`
` rArr ((1- 2sqrt(v))/(2v^(3//2))) dv = (dx)/x `
On integrating both sides w.r.t x , we get
` -v^(-1//2) - log | v| = log |x | +log C' `
` rArr (y/x)^(-1//2) + log |y/x|= - log |x| - log C'`
` rArr sqrt(x/y ) + log |y| = - log C' rArr sqrt(x/y)+ log |y| = C`