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Having many problems to understand differential equations, I've found useful an answer Sophie Swett gave me here, dispatching equations in different kinds:

  • An "ordinary" equation has variables which stand for numbers, and a solution to the equation is a number (or collection of numbers) which makes the equation true.

  • A functional equation has variables which stand for functions, and a solution to the equation is a function (or collection of functions) which makes the equation true.

  • A differential equation is a type of functional equation. Specifically, a differential equation is a functional equation that involves a derivative.

I cannot remember a chapter among those I've learnt in middle school and high school being only about functional equations, and not linked in any manner to derivatives. Except, maybe, for inverse functions or function composition.

But I don't remember any time we had a need to search a function, for itself, for a solution, from an equation.

Where were they met?

Marc Le Bihan
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    There are no functional equations in high school exams. Only in competition maths. It makes sense. The solution methods often require either copious amounts of trial-and-error or creativity. They are not suitable for high school exams (unless the question is heavily "railroaded") – Benjamin Wang Nov 26 '24 at 11:04
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    In the U.S., functional equations are usually first seen in an advanced undergraduate (3rd-4th year of college/university) real analysis course when/if the exponential and/or trigonometric functions are defined/characterized by functional equations. Sometimes this will be in an honors level elementary calculus course, such as would use Spivak's Calculus. In fact, my recollection from the late 1970s through 1980s is that then it was common (at least in the U.S.) for many students to not have seen functional equations (continued) – Dave L. Renfro Nov 26 '24 at 11:40
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    until presented with the Cauchy functional equation (in a graduate real analysis class covering Lebesgue integration and measure theory) as having non-measurable solutions. Sometimes the Cauchy functional equation would also be seen in an advanced undergraduate real analysis course, where students would be asked to prove things like if a solution is continuous somewhere (i.e. assume continuity at a single point) then it was everywhere continuous. – Dave L. Renfro Nov 26 '24 at 11:44

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