# Math Help - A separable first-order differential equation

1. ## A separable first-order differential equation

Please, help me to solve this problem. I don't know how to deal with (x) near dy and dx...

2. $
\begin{gathered}
ye^y dy = \frac{1}
{x}dx \hfill \\
\int {ye^y dy = ye^y - \int {e^y dy} = e^y \left( {y - 1} \right) = \ln x} + C \hfill \\
\end{gathered}
$

3. One more step just for fun:

$\frac{1}{e}\left(e^y(y-1)\right)=(ln(x)+c)\frac{1}{e}$

$(y-1)e^{y-1}=\frac{1}{e}(ln(x)+c)$

Taking the Lambert-W function of both sides:

$y-1=\textbf{W}\left[\frac{1}{e}(ln(x)+c)\right]$

or:

$y(x)=1+\textbf{W}\left[\frac{1}{e}(ln(x)+c)\right]$