Question

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

Hint:

This is variable-separable; both sides integrate to an arctangent.

Answer 1

Step 1: Confirm the equation is separable:
The right side factors as a function of $x$ alone times a function of $y$ alone (here, purely a ratio), so divide both sides by $(1+y^2)$ and multiply by $dx$.

Step 2: Integrate using the standard arctan antiderivative:
$\int\frac{dt}{1+t^2}=\tan^{-1}t+C$ applies to both sides, giving $\tan^{-1}y=\tan^{-1}x+C$.

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