## Differential equation selection and linear transformations

This may be vague, so I apologize.

I am interested in applied mathematics, so my question is about the process a scientist or engineer uses to determine what differential equation to use for a non-linear process. I am not familiar enough with describing non-linear processes to be able to give you an example, but from what I hear, nonlinear processes are everywhere around us. I have also read that a linear process's inputs are proportional to their outputs, and that they follow the superposition rule.

So, If I were to collect some experimental values on something like testing the height of a wave produced by different sized boats, I could develop a graph of height vs boat hull size. How would I determine if the data collected were linear or nonlinear? If nonlinear, regardless of the differential equation used, how would I transform this into a linear set of equations. So, my question is what constraints have to be met to make a nonlinear process linear? And what effect, does the number of variables play in this analysis?
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 First, you should make a distinction between data analysis and modeling. If you have a set of data points like wave height vs boat hull size, you can plot it and try to find a function, linear or otherwise, that fits your data. This already gives you some idea about the complexity of the problem, but doesn't really give you a lot of physical insight. When you know more about the physical process governing the data, you can construct the differential equation. Some physical processes are linear (take a look at the wiki page for derivation of the wave equation), and some are nonlinear (like navier stokes), but even solutions of simple linear equations do not look linear. dy/dt=y for instance has y=A*exp(x) as a solution. The solution is nonlinear, but the equation is. When you end up with a nonlinear equation, you might be able to transform it to a linear equation and there are several ways of doing it. Most of these methods are connected somehow, but Cartan's equivalence method is one of them, as well as symmetry mappings. I'm not sure if this is what you're after.. Anyway, I don't really know what you're really after because as you said yourself, your question is rather broad/vague. If you are more specific, we can be too.

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