Question

If $\sin\left(\frac{y}{x}\right) = \log_e |x| + \frac{\alpha}{2}$ is the solution of the differential equation $x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x$and $y(1) = \frac{\pi}{3}$, then $\alpha^2$ is equal to

• A. 3
• B. 12
• C. 4
• D. 9

Answer 1

Answer: (A)

3

Answer 2

Starting with the differential equation:
$x \cos \left( \frac{y}{x} \right) \frac{dy}{dx} = y \cos \left( \frac{y}{x} \right) + x$

Step 1. Divide both sides by $x^2 \cos \left( \frac{y}{x} \right)$:
$\cos \left( \frac{y}{x} \right) \left( \frac{y}{x} \frac{dy}{dx} - \frac{y}{x^2} \right) = \frac{1}{x}$

Step 2. Let $\frac{y}{x} = t$, then $y = tx$ and $\frac{dy}{dx} = t + x \frac{dt}{dx}$, substituting into the equation:
$\cos t \left( \frac{dt}{dx} \right) = \frac{1}{x}$

Step 3. Integrate both sides:
$\sin t = \ln |x| + c$
$\sin \frac{y}{x} = \ln |x| + c$
Step 4. Using the initial condition $y(1) = \frac{\sqrt{3}}{2}$, we find $c = \frac{\sqrt{3}}{2}$.
Thus, $\alpha = \sqrt{3} \implies \alpha^2 = 3$
The Correct Answer is: 3