Let x = x(y) be the solution of the differential equation $2{\mathrm{ye}}^{\mathrm{x}/{\mathrm{y}}^2}\mathrm{dx}+\left({\mathrm{y}}^2-4{\mathrm{xe}}^{\mathrm{x}/{\mathrm{y}}^2}\right)\mathrm{dy}=0$ such that x(1) = 0. Then, x(e) is equal to :

$\begin{array}{l}2{\mathrm{ye}}^{\mathrm{x}/{\mathrm{y}}^2}\mathrm{dx}+\left({\mathrm{y}}^2-4{\mathrm{xe}}^{\mathrm{x}/{\mathrm{y}}^2}\right)\mathrm{dy}=0\\ 2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}[\mathrm{ydx}-2\mathrm{xdy}]+{\mathrm{y}}^2\mathrm{dy}=0\\ 2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}\left[\frac{{\mathrm{y}}^2\mathrm{dx}-\mathrm{x}\cdot \left(2\mathrm{y}\right)\mathrm{dy}}{\mathrm{y}}\right]+{\mathrm{y}}^2\mathrm{dy}=0\end{array}$

Divide by ${\mathrm{y}}^3$

$\begin{array}{l}2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}\left[\frac{{\mathrm{y}}^2\mathrm{dx}-\mathrm{x}\cdot \left(2\mathrm{y}\right)\mathrm{dy}}{{\mathrm{y}}^4}\right]+\frac{1}{\mathrm{y}}\mathrm{dy}=0\\ 2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}\mathrm{d}\left(\frac{\mathrm{x}}{{\mathrm{y}}^2}\right)+\frac{1}{\mathrm{y}}\mathrm{dy}=0\end{array}$

Integrating

$\begin{array}{l} \\ 2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}+\mathrm{\ell ny}+\mathrm{k}=0\end{array}$

it passes through  (0, 1)

$\Rightarrow 2{\mathrm{e}}^0+\mathrm{\ell n}1+\mathrm{k}=0\Rightarrow \mathrm{k}=-2$

Required curve : $2{\mathrm{e}}^{\mathrm{x}/{\mathrm{y}}^2}+\mathcal{l}\mathrm{n} \mathrm{y}-2=0$

For x (e) 

$2{\mathrm{e}}^{\mathrm{x}/{\mathrm{e}}^2}+\ln \mathrm{e}-2=0\Rightarrow \mathrm{x}=-{\mathrm{e}}^2{\log }_{\mathrm{e}}2$