Let $x = x(y)$ be the solution of the differential equation $y = \left(x-y\dfrac{dx}{dy}\right)\sin\!\left(\dfrac{x}{y}\right)$, $y>0$ and $x(1) = \dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to:
Step-by-Step Solution
Key Concept: Rewrite as $\dfrac{dy}{y} = \sin\!\left(\tfrac{x}{y}\right)d\!\left(-\tfrac{x}{y}\right)$, leading to $\ln y = \cos\!\left(\tfrac{x}{y}\right)+C$; use double-angle identity at $y=2$ to find $\cos x$.
Rewrite: $\dfrac{dy}{y} = \sin\!\left(\frac{x}{y}\right)d\!\left(-\frac{x}{y}\right)$.
Integrating: $\ln y = \cos\!\left(\frac{x}{y}\right)+C$.
At $x(1)=\pi/2$: $\ln 1 = \cos(\pi/2)+C=0 \Rightarrow C=0$.
So $\ln y = \cos\!\left(\tfrac{x}{y}\right)$.
At $y=2$: $\ln 2 = \cos\!\left(\tfrac{x}{2}\right)$.
$\cos x = 2\cos^2\!\left(\tfrac{x}{2}\right)-1 = 2(\ln 2)^2-1$.
Correct Answer: 4