Let y = y(x)$y = y(x)$ be the solution of the differential equation (dy)/(dx) = (( x) + y)/( x( x - x x)), x in (0, (π)/(2))$\frac{dy}{dx} = \frac{(\tan x) + y}{\sin x(\sec x - \sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$ satisfying the condition y((π)/(4)) = 2$y\left(\frac{\pi}{4}\right) = 2$. Then, y((π)/(3))$y\left(\frac{\pi}{3}\right)$ is

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