\begin{equation*}
u_{xy} = 2u_x
\end{equation*}
Let $v = u_x$:
\begin{equation*}
v_y = 2v
\end{equation*}
Solving this equation:
\begin{gather*}
\frac{d v}{d y} = 2v\\
\implies \int{\frac{d v}{v}}
= \int{2 \, d y} \\
\implies ln|v| = 2y + \phi(x)\\
\implies v = \pm e^{\phi(x)}e^{2y}\\
\implies v = \varphi'(x) e^{2y}\\
\end{gather*}
Plugging $u_x$ back in for $v$:
\begin{gather*}
\implies u_x = \varphi'(x) e^{2y} \\
\implies u = \varphi(x) e^{2y} + \psi(y)
\end{gather*}
