Question

Let \( x = x(y) \) be the solution of the differential equation: \[ y = \left( x - y \frac{dx}{dy} \right) \sin\left( \frac{x}{y} \right), \, y > 0 \, \text{and} \, x(1) = \frac{\pi}{2}. \] Then \( \cos(x(2)) \) is equal to:

• A. \( 1 - 2(\log 2)^2 \)
• B. \( 2(\log 2)^2 - 1 \)
• C. \( 2(\log 2) - 1 \)
• D. \( 1 - 2(\log 2) \)

Hint:

When solving differential equations involving trigonometric functions, ensure proper rearrangement and application of initial conditions to evaluate the unknowns at specific points.

Answer 1

The Correct Option is B

We are given the differential equation: \[ y = \left( x - y \frac{dx}{dy} \right) \sin\left( \frac{x}{y} \right), \, y > 0 \, \text{and} \, x(1) = \frac{\pi}{2}. \] Rearrange the equation to solve for \( \frac{dx}{dy} \): \[ y = x \sin\left( \frac{x}{y} \right) - y \frac{dx}{dy} \sin\left( \frac{x}{y} \right), \] \[ y + y \frac{dx}{dy} \sin\left( \frac{x}{y} \right) = x \sin\left( \frac{x}{y} \right), \] \[ \frac{dx}{dy} \sin\left( \frac{x}{y} \right) = \frac{x}{y} \sin\left( \frac{x}{y} \right) - 1. \] Now, simplify and integrate the equation. Applying the initial condition \( x(1) = \frac{\pi}{2} \), we solve for \( \cos(x(2)) \). After the solution and substitution, we find that: \[ \cos(x(2)) = 2(\log 2)^2 - 1. \]