Question

Find the general solution of the differential equation \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \]

Hint:

For separable differential equations, separate the variables and integrate both sides to find the solution.

Answer 1

Step 1: Separate variables.
We can separate the variables \( x \) and \( y \) to integrate both sides: \[ \frac{dy}{1 + y^2} = \frac{dx}{1 + x^2}. \]

Step 2: Integrate both sides.
The integral of \( \frac{1}{1 + y^2} \) is \( \tan^{-1}(y) \), and the integral of \( \frac{1}{1 + x^2} \) is \( \tan^{-1}(x) \), so we get: \[ \tan^{-1}(y) = \tan^{-1}(x) + C, \] where \( C \) is the constant of integration.

Step 3: Solve for \( y \).
Thus, the general solution is: \[ y = \tan(\tan^{-1}(x) + C). \]

Step 4: Conclusion.
The general solution to the differential equation is: \[ y = \tan(\tan^{-1}(x) + C). \]