Let y = y(x)$y = y(x)$ be the solution of the differential equation x (dy)/(dx) - y = x² x, x in (0, π)$x \frac{dy}{dx} - y = x^{2} \cot x, x \in (0, \pi)$. If y((π)/(2))=(π)/(2)$y\left(\frac{\pi}{2}\right)=\frac{\pi}{2}$, then 6y((π)/(6))-8y((π)/(4))$6y\left(\frac{\pi}{6}\right)-8y\left(\frac{\pi}{4}\right)$ 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