Variable Separable

Separation of variables works whenever the derivative factors into a product of a function of y alone and a function of x alone, allowing each side to be integrated independently. Rearranging gives \frac{dy}{1+y^2} = \frac{dx}{1+x^2}, where each side is a standard arctangent form. Integrating both sides yields \arctan y = \arctan x + C, where C is the constant of integration determined by the initial data. Applying y = 0 at x = 1 gives \arctan 0 = \arctan 1 + C, so 0 = \frac{\pi}{4} + C, hence C = -\frac{\pi}{4}. Therefore \arctan y = \arctan x - \frac{\pi}{4}, and taking the tangent of both sides gives y = \tan\left(\arctan x - \frac{\pi}{4}\right). Option y = x - 1 is the tangent-line approximation, not the exact integral curve. Option y = \tan(\arctan x) collapses to x and ignores the constant entirely. Option y = \arctan x - \frac{\pi}{4} forgets to invert the arctangent on the left side. This matches the canonical JEE Advanced separable-equation method. As a final plausibility check, substituting x = 1 gives y = \tan(\frac{\pi}{4} - \frac{\pi}{4}) = \tan 0 = 0, recovering the initial condition exactly.