I would like to know something like what's the first PDE etc.
Could you recommend book on the history of PDE/ODE?
You can refer to these articles:
Bit of surfing yourself for things should help you out!
SIAM (The Society for Industrial and Applied Mathematics) has collected some oral interviews and other materials that have some bearing on the history of ordinary and partial differential equations:
There is also a lot of material on this topic that is related to more specialized historical studies:
History of Hydraulics by Hunter Rouse and Simon Ince
A History of the Theory of Elasticity and of the Strength of Materials ... by Isaac Todhunter
and work on the Euler equations and the Navier-Stokes equations.
If I were you I would start by looking for similar threads.. ;)