So, this year I have an ODE course. We have defined what an ODE is but we haven't done it for PDE. Our definition was simply:
Let I be an open interval. We call "an ODE from $n$-th order" an equation which has the form $F(t,x,x',x'',...,x^{(n)})=0$ where $x: I\to \Bbb R^m$ is called the "unknown" of the function $F: I\times \Bbb R^{m\cdot(n+1)}\to \Bbb R^m$.
The derivatives don't count as an unknown because we can find them after finding what function $x$ is.
In our definiton, $t$ is taken from $I$. What will be for PDE? Will it be something like $F(t,s,x,x',x'',...,x^n)=0$ ?
Of course I could find a definiton on google. But I couldn't find a PDE definitions which looks similar to ODE definition I mentioned. Like this, I can truly understand the difference.
Thank you.