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 $e^{\frac{\pi}{6}}$.