Question

If \( x = f(y) \) is the solution of the differential equation \[ (1 + y^2) + (x - 2e^{\tan^{-1}y}) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), \] with \( f(0) = 1 \), then \( f\left( \frac{1}{\sqrt{3}} \right) \) is equal to:

• A. \( e^{\frac{\pi}{3}} \)
• B. \( e^{\frac{\pi}{12}} \)
• C. \( e^{\frac{\pi}{6}} \)
• D. \( e^{\frac{\pi}{4}} \)

Hint:

When solving differential equations with separable variables: - Rearrange terms to separate \( x \) and \( y \) on opposite sides. - Integrate both sides carefully and apply any initial conditions provided to solve for constants of integration.

Answer 1

The Correct Option is C

We start by solving the differential equation. Rearranging the given equation: \[ (1 + y^2) + \left( x - 2e^{\tan^{-1}y} \right) \frac{dy}{dx} = 0. \] We separate variables and integrate to find \( f(y) \). The value of \( f\left( \frac{1}{\sqrt{3}} \right) \) is calculated after performing the integration, yielding \( e^{\frac{\pi}{6}} \). Thus, the required value is \( e^{\frac{\pi}{6}} \).