Solution of differential equation \( \displaystyle\frac{dy}{dx}=\displaystyle\frac{1+x^{2}}{1+y^{2}} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( x^{3}-y^{3}+3(x-y)+C=0 \): where \( C \) is an arbitrary constant
The equation is in variables separable form:
\( \displaystyle (1+y^{2})\,dy=(1+x^{2})\,dx \)
Integrating both sides,
\( \displaystyle y+\frac{y^{3}}{3}=x+\frac{x^{3}}{3}+c \)
Multiplying by 3,
\( 3y+y^{3}=3x+x^{3}+3c \)
\( x^{3}-y^{3}+3(x-y)+C=0,\quad C=3c \)