Question

Solve: \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \] 

Hint:

To solve separable differential equations, first separate the variables and then integrate both sides using standard integration formulas.

Answer 1

Step 1: Rearrange the equation to separate the variables: \[ \frac{dy}{1 + y^2} = \frac{dx}{1 + x^2}. \] Step 2: Integrate both sides: \[ \int \frac{1}{1 + y^2} \, dy = \int \frac{1}{1 + x^2} \, dx. \] Step 3: Recognize that the integral of \( \frac{1}{1 + u^2} \) is \( \tan^{-1}(u) \): \[ \tan^{-1}(y) = \tan^{-1}(x) + C, \] where \( C \) is the constant of integration. Thus, the solution is: \[ \tan^{-1}(y) = \tan^{-1}(x) + C. \]