If y = y(x)$y = y(x)$ satisfies the differential equation 16 ( x + 9√(x)) (4 + 9 + √(x)) y dy = (1 + 2 y) dx$16 \left(\sqrt{x + 9\sqrt{x}}\right) \left(4 + \sqrt{9 + \sqrt{x}}\right) \cos y \, dy = (1 + 2 \sin y) dx$, x > 0$x > 0$ and y(256) = (π)/(2)$y(256) = \frac{\pi}{2}$, y(49) = α$y(49) = \alpha$, then 2 α$2 \sin \alpha$ is equal to:

Core Logic

Step 1: Integrating Factor

This is an LDE of form (dy)/(dx) + Py = Q$\frac{dy}{dx} + Py = Q$.

Let 2x = t 2dx = dt$2x = t \implies 2dx = dt$.

Step 2: Solution of LDE

Let x = t ² x dx = dt$\tan x = t \implies \sec^2 x dx = dt$.

Step 3: Boundary Value

Given y(π/4) = 2$y(\pi/4) = 2$:

Thus, y = x (ln| x| + 2)$y = \tan x (\ln|\tan x| + 2)$. At x = π/3$x = \pi/3$:

Chapter Mix

Class 12 Maths: Differential Equations