JEE Main 2024 (Online) 31st January Morning ShiftQuestion 1
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)}$, $x\in\left(0,\frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)=2$. Then, $y\left(\frac{\pi}{3}\right)$ is
Differential Equation
Solution
Concept. The denominator simplifies to $\sin x\cos x$; then the equation is linear with IF $\cot x$. Formulas used. (1) $\sec x-\sin x\tan x=\dfrac{1-\sin^2x}{\cos x}=\cos x$(2) $\int\dfrac{dx}{\sin x\cos x}=\ln\tan x$Step 1 - Linear form. By (1), $$y'-\frac{y}{\sin x\cos x}=\frac{\tan x}{\sin x\cos x}=\sec^2x.$$ Step 2 - IF. By (2), IF $=e^{-\ln\tan x}=\cot x$: $$(y\cot x)'=\sec^2x\cot x=\frac{1}{\sin x\cos x}\ \Longrightarrow\ y\cot x=\ln\tan x+C.$$ Step 3 - Constant. $y\left(\frac\pi4\right)=2$: $2=0+C$. So $y=\tan x\,(\ln\tan x+2)$. Step 4 - At $\frac\pi3$. $y=\sqrt3\left(2+\ln\sqrt3\right)$. Answer: (C)