`(dy)/(dx)+xsin2y=x^(3)cos^(2)y`

Answer

`e^(x^(2))tany=1/2e^(x^(2))(x^(2)-1)+c`

Explanation

The given equation can be expressed as
`sec^(2)y(dy)/(dx)+2xtany=x^(3)`
Put `tany=z`, so that `sec^(2)y(dy)/(dx)=(dz)/(dx)`
Given equation transforms to `(dz)/(dx) +2xz+x^(3)`, which is linear in z.
`I.F. = e^(2intxdx)=e^(x^(2))`
Therefore, solution is given by
`ze^(x^(2))=intx^(3)e^(x^(2))dx+c`
or `tanye^(x^(2))=1/2e^(x^(2))(x^(2)-1)+c` (substitute for `x^(2)=t` and then integrate by parts)