If x (1+y^2 ) d x+y (1+x^2 ) d y=0 and y(0)=1 then x^2 y^2+x^2+y^2 is equal to

Step-by-step solution

Given that, aligned x (1+y^2 ) d x+y (1+x^2 ) d y & =0, y(0)=1 x 1+x^2 d x+ y 1+y^2 d y & =0 aligned On integrating both sides, we get 1 2 (1+x^2 )+ 1 2 (1+y^2 )=c Now, array rlrl & & y(0)=1 c= 1 2 (1+1) & = 1 2 2 & & & 1+x^2 1+y^2 = & & 2 & & (1+x^2 ) (1+y^2 ) & =2 & & x^2+y^2+x^2 y & =1 array