I'm trying to use MATLAB's PDE solver to solve the following PDE
$$ (-Lx + \gamma y^2 - F)u_x +(-Sy - \gamma xy)u_y + \sigma^2 u_{yy} = -1 $$
$$ L, \gamma, F, S, \sigma^2 \in R $$
I need to put this into divergence form to use with MATLAB's solver
$$ -\nabla \cdot(c \nabla u)+a u=f $$
a, c, and f are functions of position (x, y) and possibly of the solution u.
If we write
$$ c=\left(\begin{array}{ll} c_1 & c_3 \\ c_2 & c_4 \end{array}\right) $$
then,
$$ \begin{aligned} & \nabla \cdot(c \nabla u)=c_1 u_{x x}+\left(c_2+c_3\right) u_{x y}+c_4 u_{y y} \\ & +\left(\frac{\partial c_1}{\partial x}+\frac{\partial c_2}{\partial y}\right) u_x+\left(\frac{\partial c_3}{\partial x}+\frac{\partial c_4}{\partial y}\right) u_y \end{aligned} $$
Therefore $c_1 = 0, \quad c_4 = \sigma^2$
$c_2 = -Lxy + \gamma y^3/3 - F + K_2(x)$
$c_3 = -Sxy - \gamma yx^2/2 + K_3(y)$
However, this won't satisfy the constraint of $c_2 + c_3 = 0$
My expertise isn't in PDE's. Is there a trick in converting this into the required form. Or alternatively, is there a better way to solve this.
Thanks in advance.