# How are differential equations found?

after finishing school one usually has an understanding of the connection between - say - the velocity of a car and its acceleration. Concluding, it's not hard for young students to transform a question like

The car has an acceleration of $$a(t)$$, how can its velocity be described?

We need to find $$v$$ such that $$v'(t) = a(t)$$.

However I was wondering for quite some time now, how people got to the idea of writing down some of the much more complex PDE that we work with nowadays. For example things like

• the Laplace / Poisson equation
• equation of (non-)linear elasticity
• the Navier-Stokes equation
• etc.

So how did people do it? I can hardly imagine that they just conducted thousands of experiments and said 'hey wow, this must be a 5th derivative there!'.

## My guess would be the following:

We have conservation of mass, conservation of energy and conservation of whatever (at least we assume it), so for the domain $$\Omega\subset\mathbb{R}^d$$ to conserve its energy/mass/whatever we must have

$$\int_\Omega f_{in}(x)-f_{out}(x) dx= 0$$

where $$f_{in}$$ denotes incoming energy/mass/whatever and $$f_{out}$$ denotes outgoing energy/mass/whatever. Now any PDE of the above mentioned is only one or more of these conservation laws, in which we plug in constitutive relations for the specific situation, that allow us to write $$f_{in}$$ and $$f_{out}$$ explicitly.

What do you think?

I read lots of math books during my studies, but I could not find an answer so far...

Thank you !

• It's all about describing how things change over time. Can be PDE, ODE, integral equation, etc. You may want to read an introduction book about mathematical physics. This article in wiki may help to understand some concepts: en.wikipedia.org/wiki/Heat_equation Commented Oct 21, 2019 at 13:47

On one hand you have equations between physical quantities such as the fundamental law of dynamics,

$$F=ma$$ where $$F$$ is the force applied to a mobile of mass $$m$$ and $$a$$ the acceleration it takes, or

$$F=-ke$$ expressing the force exerced by a spring of stiffness $$k$$ when its endpoint is displaced by $$e$$.

Such laws are established and validated by experimentation and by reasoning (also by dimensional analysis).

On the other hand, you have differential relations such as

$$a(t)=\ddot e(t),$$ i.e. the acceleration is the second derivative of space over time.

Now you get the well-known ODE for the "harmonic oscillator":

$$m\ddot e(t)+ke(t)=0.$$

• Thanks for your answer, I understand. Your example is nice, but really easy. So also really different relations between different quantities are in their core the result of experiments. And why do so many PDE use the conservation laws? I mean, even if I e.g. know the transport theorem and whatever else I need, how do I know that in order to write the Navier-Stokes equation down I must consider mass-balance also?
– Denn
Commented Oct 21, 2019 at 13:45
• @Denn: the equations form a description of the system and constrain the unknowns. In particular, you express that a system is closed by stating that the total mass is constant.
– user65203
Commented Oct 21, 2019 at 16:37