The general solution of \(\displaystyle\frac{dy}{dx}=1+x^2+y^2+x^2y^2\) is:
(given that C is the constant of integration)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\tan^{-1}y=x+\displaystyle\frac{x^3}{3}+C\)
The right-hand side factorises:
\(1+x^2+y^2+x^2y^2=(1+x^2)+y^2(1+x^2)\)
\(=(1+x^2)(1+y^2)\)
So the variables separate:
\( \displaystyle \frac{dy}{1+y^2}=(1+x^2)\,dx \)
Integrating both sides:
\( \displaystyle \int\frac{dy}{1+y^2}=\int(1+x^2)\,dx \)
\( \displaystyle \tan^{-1}y=x+\frac{x^3}{3}+C \)