Question

(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):

Hint:

Substitution is a powerful tool for solving first-order differential equations.

Answer 1

Rewriting the equation: \[ \frac{dy}{dx} = \frac{1 + y^2}{\tan^{-1} y - x}. \] Use substitution \( z = \tan^{-1} y - x \), then differentiate and solve. The solution is: \[ z = C, \quad \text{where } z = \tan^{-1} y - x. \] Thus: \[ \tan^{-1} y - x = C. \]