Question

If \( x \sin \left( \frac{y}{x} \right) \, dy = \sin \left( \frac{y}{x} \right) - x \, dx \) and \( y(1) = \frac{\pi}{2} \), then the value of \( \cos \left( \frac{y}{x} \right) \) is

• A. \( x \)
• B. \( \frac{1}{x} \)
• C. \( \log x \)
• D. \( e^x \)

Hint:

When solving differential equations, use initial conditions to find the particular solution.

Answer 1

The Correct Option is C

Step 1: Solving the differential equation.
Integrate both sides of the equation \( x \sin \left( \frac{y}{x} \right) \, dy = \sin \left( \frac{y}{x} \right) - x \, dx \) using the given initial condition. The solution leads to \( \cos \left( \frac{y}{x} \right) = \log x \).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } \log x} \]