One reason that Black-Scholes equations works in mathematical finance is that there are only so many ways you can write an equation that describes stock prices that does not blow up.

Answer by Joseph Wang:

This is an issue not only with physics but in any situation where differential equations are used.

What you will find if you try to create an equation that uses higher order differential equations is that the equation will tend to be unstable and blow up. If you solve an equation that has higher order terms, then what ends up happening is that you will end up with solutions that are unstable in the sense that any small changes in your initial parameters will result in your final solution being wildly different. Since there is always uncertainty in your initial conditions, you end up with an equation that doesn't tell you anything.

You can find some situations where the higher order terms end up being well behaved, but invariably there is some special mathematical coincidence or special case that causes this to work. Usually you'll find some external constriant that keeps the equations from going completely out of bounds, which is why you do see these equations in some engineering contexts. But if there is nothing that forces the equation to produce numbers with limited values, they tend to scatter all over the place.

This applies not just in physics. One reason that Black-Scholes equations works in mathematical finance is that there are only so many ways you can write an equation that describes stock prices that does not blow up.

One other way of thinking about this (which is mathematically equivalent). Suppose I give you a bunch of data and you want to do a curve fit. You might do a linear fit, if it looks like there is a minima, you might want to do a quadratic fit. If you start fitting cubic and higher order polynomials, you'll likely be fitting noise, and you'll quickly find that what you have is pretty meaningless.

Why don't differential equations of physics go beyond the second order?

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