Solve the differential equation `x y(dy)/( dx)=(1+y^2)/(1+x^2)(1+x+x^2)`

Explanation

We have `(ydy)/(1+y^(2))=(1+x+x^(2))/(x(1+x^(2))`dx
Integrating both sides,
`int(ydy)/(1+y^(2))=int(1/x+1/(1+x^(2)))dx`
`therefore 1/2log_(e)(1+y^(2))=log_(e)x+tan^(-1)x+log_(e)C`
`log_(e)sqrt(1+y^(2))/(cx)=tan^(-1)x`
`rArr sqrt(1+y^(2))=cxe^(tan^(-1)x)`