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: Recognising the separable form:
The equation splits cleanly as a function of \(y\) times a function of \(x\), so cross-divide and integrate term by term rather than substituting.

Step 2: Integrating the y-side:
\(\displaystyle\int(1+y^2)^{-1}dy=\tan^{-1}y+C_1\).

Step 3: Integrating the x-side:
\(\displaystyle\int(1+x^2)\,dx=x+\dfrac{x^3}{3}+C_2\).

Step 4: Combining constants:
Equate the two antiderivatives, merging \(C_1,C_2\) into one constant \(C\).

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