If `y=x/(In|cx|)` (where c is an arbitrary constant) is the general solution of the differential equation `(dy)/(dx)=y/x+phi(x/y)` then function `phi(x/y)` is:

Options

A. `x^(2)//y^(2)`
B. `-x^(2)//y^(2)`
C. `y^(2)//x^(2)`
D. `-y^(2)//x^(2)`

Answer

D

Explanation

`logc+log|x|=x/y`
Differentiating w.r.t. `x,1/x=(y-x(dy)/(dx))/(y^(2))`
or `y^(2)/x=y-x(dy)/(dx)`
or `(dy)/(dx)=y/x-y^(2)/x^(2)`
or `phi(x/y) = -y^(2)/x^(2)`