The real task of the mathematical physicist
What exactly IS a DIFFERENTIAL EQUATION and how does it arise in the first place? Most physics courses start by taking the equation as a GIVEN – not bothering to explain how it comes about in the first place. And then, you spend a whole semester SOLVING these beasts – without knowing how they originated in the first place.
The real task, in my opinion, is to FORMULATE PHYSICS problems. This is done (typically) – using infinitesimal analysis of the problem…(Enter Calculus). This analysis in the end gives an output called a differential equation.
A differential equation specifies an unknown function, instead of a specification of the function itself in terms of values alone.
It took me a few years to understand the highlighted sentence above.
Differential equations effectively define new functions, called their solutions, and it is the task of the mathematician to elucidate their properties so that they can be used with effect and confidence. Bessel functions are of this type, solutions of certain differential equations that arise in many different connections. (Quote from JB Calvert’s notes)
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