Question

Find the general solution of \(\dfrac{dy}{dx}=(1+x^2)(1+y^2)\).

Hint:

Separate variables: dy/(1+y²) on one side, (1+x²)dx on the other, then integrate.

Answer 1

Step 1: Separating variables:
Divide both sides by \((1+y^2)\): \(\dfrac{dy}{1+y^2}=(1+x^2)\,dx\).

Step 2: Integrating both sides:
\(\displaystyle\int\dfrac{dy}{1+y^2}=\int(1+x^2)\,dx\).

Step 3: Evaluating:
\(\tan^{-1}y=x+\dfrac{x^3}{3}+C\).

Final Answer:
\[ \boxed{\tan^{-1}y=x+\dfrac{x^3}{3}+C} \]