`(dy)/(dx)=(x^(3)-2xtan^(-1)y)(1+y^(2))`

Answer

`2tan^(-1)y=x^(2)-1+2Ce^(-x^(2))`

Explanation

`1/(1+y^(2)).(dy)/(dx)+2x(tan^(-1)y)=x^(3)`………..(1)
Put `tan^(-1)y=z`
`therefore 1/(1+y^(2)).(dy)/(dx)=(dz)/(dx)`
Thus, (1) reduces to `(dz)/(dx)+(2x)z=x^(3)`, which is linear differential equation
I.F. `=e^(int2xdx)=e^(x^(2))`
Thus, solution is `z.e^(x^(2))=1/2int2e^(x^(2))-e^(x^(2))+2C`
or `2tan^(-1)y=x^(2)-1+2Ce^(-x^(2))`