Question #67732

Let $y = y(x)$ be the solution to the differential equation $\frac{d}{dx} (\sin^{-1}(\sin y)) + 2x y = 2\pi x + x^2 \int_{-\pi/4}^{\pi/4} \sin^5 t \cos^2 t \, dt$ for $x \in (0, 1)$, with the initial condition $y(0) = \frac{5\pi}{6}$. When this differential equation is reduced to the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$, let $I(x) = e^{\int P(x) dx}$ be its integrating factor.

Let $k = I(\sqrt{\ln 2})$. Consider the following system of linear equations in variables $x, y, z$: $x + 2y + 3z = 2$ $8x + 12y + 16z = 5$ $3x + 4y + 5z = k$ Let $S$ be the set of all solutions $(x,y,z)$ to this system. Which of the following statements is true?