The solution of differential equation `x^(2)(x dy + y dx) = (xy - 1)^(2) dx` is (where c is an arbitrary constant)

Options

A. `xy - 1 = cx`
B. `xy - 1 = cx^(2)`
C. `(1)/(xy-1)=(1)/(x)+c`
D. None of these

Answer

C

Explanation

Given differential equation can be written as
`(xdy+ydx)/((xy-1)^(2))=(dx)/(x^(2)) rArr (d(xy))/((xy-1)^(2))=(dx)/(x^(2))`
Integrating both sides
`-(1)/((xy-1))=-(1)/(x)+ c rArr (1)/(xy-1)=(1)/(x) + c`