How to set up an equation for a real system?

[amorale]

Hello

The question I have is how do mathematicians go about creating an equation to model a real system. I would like to know how they go about doing this, any links would be nice.

To be more clear let's say scientists are trying to study how carbonic acid and temperature affect the pH of seawater this are two independent variables so some type of differential equation must be found to be able to model the system and see how each variable independently will affect the pH, this way by using partials derivatives they can find the max and mins of the function. I think that a hypothetical equation might be f(x,y)= x^2+y^3 where f(x,y) is the pH, x is the temperature and y is the carbonic acid. How do they get to the equation?

That is a hypothetical example I came up with, to illustrate were i am coming from. Like I say I just want to know how scientist, mathematicians and engineers go about solving this real life problems to be able to model them.

Thanks in advance

P.S
I have knowledge of multivariable calculus and of ordinary differential equations.
I apologized for any spelling mistakes.

 

[chiro]

Hey amorale and welcome to the forums.

The typical way that this occurs is that you introduce assumptions that lead to mathematical constraints and then you try and model how a system changes over time.

The changes are usually reflected in things like derivatives in differential equations which may be multi-variable PDE's or ODE's, both with linear or non-linear models.

The final model selected is one that is good enough to use for purposes while not being too complicated enough to actually use and make sense of.

The actual trade-off depends on the nature of the problem, who is using it, why its being used, and and what it is used for.