If $y=y(x)$ satisfies the differential equation $16\left(\sqrt{9x+x\sqrt{x}}\right)\left(4+\sqrt{9+\sqrt{x}}\right)$ $\cos y\text{ }dy=(1+2\sin y)$$dx,\text{ }x0$ and $y(256)=\frac{\pi }{2},\text{ }y(49)=\alpha ,$ then $2\sin \alpha$ is equal to [JEE Main 2026, 22 Jan (Shift 2)]

Question

If $y=y(x)$ satisfies the differential equation $16\left(\sqrt{9x+x\sqrt{x}}\right)\left(4+\sqrt{9+\sqrt{x}}\right)$ $\cos y\text{ }dy=(1+2\sin y)$$dx,\text{ }x>0$ and $y(256)=\frac{\pi }{2},\text{ }y(49)=\alpha ,$ then $2\sin \alpha$ is equal to [JEE Main 2026, 22 Jan (Shift 2)]

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If \(y=y(x)\) satisfies the differential equation \(16\left(\sqrt{9x+x\sqrt{x}}\right)\left(4+\sqrt{9+\sqrt{x}}\right)\)…

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