If ((y)/(x)) = ₑ|x| + (α)/(2)$\sin \left(\frac{y}{x}\right) = \log_{e}|x| + \frac{\alpha}{2}$ is the solution of the differential equation x ((y)/(x))(dy)/(dx) = y ((y)/(x)) + x$x\cos \left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos \left(\frac{y}{x}\right) + x$ and y(1) = (π)/(3)$y(1) = \frac{\pi}{3}$, then α²$\alpha^2$ 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