Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& \left(1+y^2\right)e^{\tan x}dx+{\cos }^{2}x\left(1+e^{2\tan x}\right)dy=0\\ & \int \frac{{\sec }^{2}x{e}^{\tan x}}{1+e^{2\tan x}}dx+\int \frac{dy}{1+y^2}=C\\ & \Rightarrow {\tan }^{-1}\left(e^{\tan x}\right)+{\tan }^{-1}y=C\\ & \text{ for }x=0,y=1,{\tan }^{-1}(1)+{\tan }^{-1}1=C\end{array}$ $\mathrm{C}=\frac{\pi }{2}$ ${\tan }^{-1}\left(e^{\tan x}\right)+{\tan }^{-1}y=\frac{\pi }{2}$
Put $x=\pi ,{\tan }^{-1}e+{\tan }^{-1}y=\frac{\pi }{2}$ ${\tan }^{-1}y={\cot }^{-1}e$ $y=\frac{1}{e}$