Find the general solutions to
\begin{equation}
u_{xxyy}=\sin(x)\sin(y);
\label{eq1}
\end{equation}
Solution:
\begin{equation}
u_{xxy}=-\sin(x)\cos(y)+\phi(x);
\label{eq2}
\end{equation}
\begin{equation}
u_{xx}=-\sin(x)\sin(y)+y\phi(x)+\psi(x);
\label{eq3}
\end{equation}
\begin{equation}
u_{x}=\cos(x)\sin(y)+y\int \phi(x)\,dx+\int\psi(x)\,dx+h(y)=\cos(x)\sin(y)+yf(x)+g(x)+h(y);
\label{eq4}
\end{equation}
\begin{equation}
u=\sin(x)\sin(y)+y\int f(x)\,dx+\int g(x)\,dx+xh(y)+\eta(y)=\sin(x)\sin(y)+yF(x)+G(x)+xh(y)+\eta(y）;
\label{eq5}
\end{equation}
