The solution of differential equation
`(x+y(dy)/(dx))/(y-x(dy)/(dx)) = x(cos^(2)(x^(2)+y^(2))/(y^(3)))` is

Options

A. `tan(x^(2)+y^(2))= x^(2)/y^(2)+c`
B. `cot(x^(2)+y^(2))=x^(2)/y^(2)+c`
C. `tan(x^(2)+y^(2))=y^(2)/x^(2)+c`
D. `cot(x^(2)+y^(2))=y^(2)/x^(2)+c`

Answer

A

Explanation

The given equation can be written as
`(xdx+ydy)/((ydx-xdy)//y^(2))=y^(2)x/y^(3)cos^(2)(x^(2)+y^(2))`
or `(xdx+ydy)/(cos^(2)(x^(2)+y^(2)))=x/y(ydx-xdy)/(y^(2))`
`=1/2sec^(2)(x^(2)+y^(2))d(x^(2)+y^(2))=x/yd(x/y)`
On integrating, we get
`1/2tan(x^(2)+y^(2))=1/2(x/y)^(2)+c/2`
or `tan(x^(2)+y^(2))+c`