The solution of the differential equation
`{1/x-y^(2)/(x-y)^(2)}dx+{x^(2)/(x-y)^(2)-1/y}dx=0` is

Options

A. ` In |x/y|+(xy)/((x-y))=C`
B. ` In |xy|+(xy)/((x-y))=C`
C. ` (xy)/((x-y))=Ce^(x//y)`
D. ` (xy)/((x-y))=Ce^(xy)`

Answer

a

Explanation

The given equation can be written as
` ((dx)/y-(dy)/y )+(x^(2)dy-y^(2)dx)/((x-y)^(2))=0`
` rArr ((dx)/x - (dy)/y)+(((dy)/(y^(2)) -(dx)/(x^(2))))/ ((1/y-1/x)^(2))=0`
` rArr ((dx)/x - (dy)/y)+(((dy)/(y^(2)) -(dx)/(x^(2))))/ ((1/x-1/y)^(2))=0`
On integrating both sides , we get
` In |x| - In |y| - 1/((1/x-1/y))=C`
` rArr In |x/y| = (xy)/((y-x))= C rArr In |x/y| +(xy)/((x-y)) = C`