I'm told that the PDE
$$\frac{\partial^2u(s, t)}{\partial{s}\partial{t}} = -\frac{1}{\sigma^2}f(s,t)$$
can be solved by direct integration to get
$$u(s, t) = F(s) + G(t) - \phi(s, t)$$
where $F(s)$ and $G(t)$ are arbitrary functions and
$$\phi(s, t) = \frac{1}{\sigma^2} \iint f(x(s, t), y(s, t)) \ ds \ dt$$
Can someone please demonstrate the steps for how this was done?
The reason I am confused is because I had another integral (see this question), and it was solved using the fundamental theorem of calculus, as follows:
$$\dfrac{du}{dt} = f$$ $$\therefore \int_0^t \frac{du}{dt'} \ dt' = \int_0^t f(x(s, t'), y(s, t')) \ dt'$$ $$\therefore u(s, t) = u(s, 0) + F(t), F(t) = \int_0^t f(x(s, t'), y(s, t')) \ dt'$$
However, it doesn't seem like the former integral uses the FTC? I'm confused as to why it's used in one case and not the other?
Thanks to all who kindly take the time to explain this.