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: Separate the variables:
\[ \dfrac{dy}{1+y^2}=\dfrac{dx}{1+x^2} \]

Step 2: Integrate both sides:
\[ \int\dfrac{dy}{1+y^2}=\int\dfrac{dx}{1+x^2} \]
\[ \tan^{-1}y=\tan^{-1}x+C \]

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