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

Options

A. `"ln"|x/y|+(xy)/(x-y)=c`
B. `(xy)/(x-y)=ce^(x//y)`
C. `"ln"|xy|+(x^(4)y^(4))/4=C`
D. None of these

Answer

A

Explanation

`{1/x-y^(2)/(x-y)^(2)}dx+{x^(2)/(x-y)^(2)-1/y}dy=0`
or `((dx)/x-(dy)/y)+((x^(2)dy-y^(2)dx))/((x-y)^(2))=0`
or `((dx)/x-(dy)/y)+((dy//y^(2)-dx//x^(2))/((1//y-1//x)^(2)))=0`
Integrating, we get `"ln "|x|-"ln "|y|-1/(1//x-1//y)=c`
or `"ln "|x/y|+(xy)/(x-y)=c`