Note that I am not a mathematician; I am simply deducing using the very fallible means of deduction via intuition, which by no means is rigorous.
My question concerns the possibility for any PDE to be written as an integral equation. For example, suppose we have the differential equation:
$$ \frac{\partial f}{\partial x} = 3xy $$
It is clear that integrating would yield a solution (not the general solution, but a solution) to the differential equation in this case:
$$ f(x, y) = \int 3xy~dx = \frac{3x^2 y}{2} $$
My reasoning is that given a certain PDE in the form:
$$ \frac{\partial f}{\partial x} = g(x, y) $$
It should be able to be recast into the form:
$$ f(x, y) = \int g(x, y)~dx = \int \limits_0^x g(t, y)~dt $$
And given the linearity of both the partial derivative and the integral with respect to one variable, shouldn't it be possible to write any arbitrary PDE composed of a finite number of sums and differences of a partial derivatives of a function as an integral equation? For instance, given:
$$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z} = g(x, y, z) $$
Wouldn't it be possible to rewrite it as:
$$ f(x, y, z) = \frac{1}{3} \left[ \int \limits_0^x g(t, y, z)~dt + \int \limits_0^y g(x, u, z)~du + \int \limits_0^z g(x, y, v)~dv \right] $$
I already have the feeling that I am operating from gruesome oversimplification, and once again, I have no rigorous proof, but would this be true?