I am trying to solve for the function $A(x,y)$ given an equation of the form
$\frac{\partial}{\partial y}\int_{x_0}^xA(x,y)f_1(x,y)dx + \frac{\partial}{\partial x}\int_{y_0}^yA(x,y)f_2(x,y)dy = A(x,y)f_3(x,y) \tag{1}$
where $f_1$, $f_2$, and $f_3$ are known differentiable/integrable functions on the plane. (I will generally have algebraic expressions for these three functions, which are informed by the physical interpretation of the problem.) I'm interested in either obtaining a general solution, or figuring out an approach to solving, since the functions $f_i$ will differ between cases where I apply this. The most appropriate boundary conditions are constants; for example, $A(x, y_0) = A_1$ and $A(x_0, y) = A_2$, where $A_1$ and $A_2$ are real. I would prefer to be able to solve analytically, but I can resort to numerical approximations if necessary.
I've noticed that I can differentiate Equation 1 once with respect to $x$ and once with respect to $y$ to obtain something of the form
$\partial_{yy}(Af_1) + \partial_{xx}(Af_2) = \partial_{xy}(Af_3) \tag{2}$
which can be expanded out, using the product rule, to become
$f_2\partial_{xx}A - f_3\partial_{xy}A + f_1\partial_{yy}A+ (2\partial_xf_2 - \partial_yf_3)\partial_xA + (2\partial_yf_1 - \partial_xf_3)\partial_yA + (\partial_{xx}f_2 + \partial_{yy}f_1 - \partial_{xy}f_3)A = 0. \tag{3}$.
I've looked up this form and this seems to be a second-order linear PDE with variable coefficients. Depending on location and circumstance, it can be either elliptic or hyperbolic, and sometimes (rarely) parabolic. The situation I am the most interested in is where the equation is elliptic. Can this PDE be solved analytically to yield an expression for $A$ that satisfies Equation 1? Or are there numerical methods I could apply to solve Equation 1?
To be a little more concrete: in the simplest-case scenario of what I'd like to solve, Equation 3 (as written above) becomes
$-2y^2\partial_{xx}A +12y(x - x_0)\partial_{xy}A + (y^2 -3[x - x_0]^2 + K)\partial_{yy}A + 12(x - x_0)\partial_xA + 16y\partial_yA +14A = 0$
with $K$ a known constant.