The solution of the differential ` x(x-y) (dy)/(dx) = y(x+y)`, is

Options

A. ` x/y + log(xy)=c`
B. `y/x+ (log (xy)=0`
C. ` x/y+ y log x =c`
D. `x/y + x log y=c`

Answer

A

Explanation

Given ` , (dy)/(dx) = (y (x+y))/(x(x-y))`
put y = vx
` Rightarrow (dy)/(dx) = v + (dv)/(dx)`
`v + x(dv)/(dx) = (vx (x + vx))/(x(x-vx))`
` Rightarrow x(dv)/(dx) = (2v^(2))/(1-v)`
` Rightarrow 1/2 [ 1/v^(2) -1/v] dv = (dx)/x`
on integrating , we get
` 1/2 [ - 1/v - log v] = log x + c_(1)`
` Rightarrow x/y + log (y/x) + 2log x = 2-c_(1)`
` Rightarrow x/y +log(xy)=c`