The solution of the differential equation
`(dy)/(dx) = 1/(xy[x^(2)siny^(2)+1])` is

Options

A. `x^(2)(cosy^(2)-siny^(2)-2Ce^(-y^(2)))=2`
B. `y^(2)(cosx^(2)-siny^(2)-2Ce^(-y^(2)))=4C`
C. None of these
D. a system of circles

Answer

A

Explanation

`(dy)/(dx) = 1/(xy[x^(2)siny^(2)+1])`
or `1/x^(3)(dx)/(dy) -1/x^(2)y=ysiny^(2)`
Putting `-1//x^(2)=u`, we get
`(du)/(dy)+2uy=2ysiny^(2)`.
I.F. `=e^(y^(2))`
Thus, solution is `ue^(y^(3))=int2ysiny^(2)e^(y^(2))dy+C`
`=int(sint)e^(t)dt+C`
`=1/2e^(y^(3))(siny^(2)-cosy^(2))+c`
or `2u=(siny^(2)-cosy^(2))+2Ce^(-y^(2))`
or `2=x^(2)[cosy^(2)-siny^(2)-2Ce^(-y^(2))]`