Question

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

Hint:

Separate variables as \(\dfrac{dy}{1+y^2}=\dfrac{dx}{1+x^2}\) and integrate both sides.

Answer 1

Step 1: Understanding the Concept:
This equation is variable separable: all the \(y\) terms can be gathered on one side and all the \(x\) terms on the other.

Step 2: Separating the variables:
\(\dfrac{dy}{1+y^2}=\dfrac{dx}{1+x^2}\).

Step 3: Integrating both sides:
\(\displaystyle\int\dfrac{dy}{1+y^2}=\int\dfrac{dx}{1+x^2}\ \Rightarrow\ \tan^{-1}y=\tan^{-1}x+C\).

Final Answer:
General solution: \(\boxed{\tan^{-1}y=\tan^{-1}x+C}\).