# Proving compatibility of two Partial differential equations

Given two PDE(s): $F(x,y,z,p,q)=0$
and $G(x,y,z,p,q)=0$
In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of $G=0$,then $F=0$ and $G=0$ are said to be compatible. But according to another textbook if $F=0$ and $G=0$ have atleast one common solution then they are said to be compatible.So here my question is:
What is the actual "definition of compatibility of two pde(s)"?

Now if I assume that $\left|\frac{\partial(F,G)}{\partial(p,q)}\right|\ne0$ and solve the system $F=0$ and $G=0$ for $p$ and $q$,suppose that I get $p=\phi(x,y,z)$ and $q=\psi(x,y,z)$ so writing complete differential for $dz$,
$dz=pdx +qdy$ ,so this Pfaffian equation is integrable?(This condition has been used in textbook to deduce the condition for two partial differential equations to be compatible).
My second question is:
How does compatibility of "$F=0$ and $G=0$" plus the condition $\left|\frac{\partial(F,G)}{\partial(p,q)}\right|\ne0$ imply integrability of Pfaffian equation "$dz=pdx +qdy$" ?
PS:$p=z_x,q=z_y$
Your third condition, which is that, when one solves for $p$ and $q$ as functions of $x$, $y$, and $z$ and substitutes these into the $1$-form $\theta = \mathrm{d}z - p\,\mathrm{d}x - q\,\mathrm{d}y$, one obtains an integrable $1$-form on $xyz$-space, i.e., one that satisfies that $\theta \wedge \mathrm{d}\theta$ vanishes identically, is the standard definition of compatibility of the pair of equations. What it ensures is that there is a nontrivial $1$-parameter family of (local) functions $z(x,y,c)$ (where $c$ is the nontrivial parameter) that simultaneously satisfy both equations.
Note that this is properly stronger than the condition that there exists a single function $z(x,y)$ that satisfies both equations, and it does not imply that every (local) solution of one of the equations is a (local) solution of the other.