Solve the differential equation `(1 + x^(2))dy+2xydx = sin^(2) x dx`

Explanation

`(dy)/(dx) + (2x)/(1+x^(2))y = (sin^(2)x)/(1 + x^(2))` ...(1)
Equation (1) is the linear differential equation of the type `(dy)/(dx) + Py = Q`, where `P = (2x)/(1+x^(2))` and `Q = (sin^(2)x)/(1+x^(2))`
`:. I.F. = e^(int(2x)/(1+x^(2))dx)`
`= e^(ln(1+x^(2)))`
`= 1 + x^(2)`
Thus, the solution of (1) is
`y(1+x^(2)) = int (sin^(2)x)/(1+x^(2))(1+x^(2))dx + c`
`= (1)/(2)int(1-cos 2x)dx + c`
`rArr y(1+x^(2)) = (1)/(2)(x-(sin 2x)/(2))+c`