The particular solution of the differential equation $y\left(\frac{\mathrm{d} x}{\mathrm{~d} y}\right)=x…
The particular solution of the differential equation $y\left(\frac{\mathrm{d} x}{\mathrm{~d} y}\right)=x \log x$ at $x=\mathrm{e}$ and $\mathrm{y}=1$ is
$\mathrm{e}^{x y}=2$
$x=\mathrm{e}^{\mathrm{y}}$
$x\mathrm{y}=2$
$\log x=2 y$
Solution
$y\left(\frac{d x}{d y}\right)=x \cdot \log x$
$\therefore \int \frac{1}{x \cdot \log x} d x=\int \frac{1}{y} d y$
$\therefore \log |\log x|=\log y+\log c$
We have $x=e$ and $y=1$
$\therefore \log |\log e|=\log 1+\log c \Rightarrow \log c=0$
$\therefore \log |\log x|=\log y \Rightarrow \log x=y \Rightarrow x=e^{y}$