Differential Equations
Linear Differential Equations
JEE Advanced 2015
Grade 12
Question:
Let $y(x)$ satisfy $\frac{dy}{dx} + 2y \tan x = \sin x$ with $y\left(\frac{\pi}{3}\right) = 0$. Then $y\left(\frac{\pi}{4}\right)$ equals:
(1) $\frac{2 - \sqrt{2}}{2}$
(2) $\frac{\sqrt{2} - 2}{2}$
(3) $-\frac{\sqrt{2} + 2}{2}$
(4) $\frac{\sqrt{2} - 1}{2}$
Step-by-Step Solution
Key Concept: Recognise this as a linear differential equation in $y$, find its integrating factor $e^{\int 2 \tan x \, dx} = \sec^2 x$, and write the solution in the form $y \sec^2 x = \int \sin x \sec^2 x \, dx$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)