Question

Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]

Hint:

For separable differential equations, express variables separately before integrating.

Answer 1

Step 1: Rewrite in separable form. \[ (1 + y^2) dy = (1 + x^2) dx \] Step 2: Integrate both sides. \[ \int (1 + y^2) dy = \int (1 + x^2) dx \] Step 3: Solve the integrals. \[ y + \frac{y^3}{3} = x + \frac{x^3}{3} + C \] Rewriting in inverse tangent form, \[ \tan^{-1} y = \tan^{-1} x + C. \]