I recently came across a system of PDEs
$\frac{\partial S}{\partial z}= f\_1(x,y,z,w,t)$,
$\frac{\partial S}{\partial w}= f\_2(x,y,z,w,t)$,
$\frac{\partial S}{\partial t}= f\_3(x,y,z,w,t)$,
$S(x,y,1,1,1)=f\_4(x,y)$,
where $S$ is an unknown function of five variables $x,y,z,w,t$ and $f\_i$ are known.
The question is how to obtain a general solution for $S$?



Solution for a  system of PDEs
What you have is a family of PDEs labelled by $x,y$ and for fixed $x,y$ you have an equation 
$$dS = f\_1 dz + f\_2 dw + f\_3 dt$$
for a function $S:\mathbb{R}^3 \to \mathbb{R}$ of the three variables $z,w,t$. The first thing to check is that the equation is integrable: namely, that the 2-form $d(dS) = 0$. If that is the case, then $S$ exists up to a locally constant function, which you fix from the fourth condition.


To find the explicit solution, choose any path $\gamma$ from $(1,1,1)$ to $(z,w,t)$ and integrate the right-hand side of the equation:
$$S(x,y,z,w,t) = S(x,y,1,1,1) + \int\_\gamma (f\_1 dz + f\_2 dw + f\_3 dt).$$


