Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0, y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to [JEE Main 2024, 8 Apr (Shift 1)]

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YESPYQ · Accepted Answer

Correct answer: d) $\frac{1}{e}$. $\left(1+{y}^{2}\right){e}^{\tan x}dx+{\cos }^{2}x\left(1+{e}^{2\tan x}\right)dy=0$ $\int \frac{{\sec }^{2}x{e}^{\tan x}}{1+{e}^{2\tan x}}dx+\int \frac{dy}{1+{y}^{2}}=C$ $\Rightarrow {\tan }^{-1}\left({e}^{\tan x}\right)+{\tan }^{-1}y=C$ $\text{for }x=0,y=1,{\tan }^{-1}(1)+{\tan }^{-1}1=C$ $C=\frac{\pi }{2}$ ${\tan }^{-1}\left({e}^{\tan x}\right)+{\tan }^{-1}y=\frac{\pi }{2}$ $\text{Put }x=\pi ,{\tan }^{-1}e+{\tan }^{-1}y=\frac{\pi }{2}$ ${\tan }^{-1}y={\cot }^{-1}e$ $y=\frac{1}{e}$

Let \(y=y(x)\) be the solution of the differential equation \(\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \t…

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