Maxwell was very good at math. However he came in only second ("Second Wrangler") in the "Tripos", the final mathematics exams at Cambridge, while someone less able, but better at cramming, was "Senior Wrangler." This prompted Maxwell's friend P.G. Tait to comment in later life, "I could teach a coal scuttle to be Senior Wrangler."
Since I am not good at math, it was not as obvious to me as it was to Maxwell that the exponential was necessarily the only function that could solve his problem. For reassurance I turned to Prof. Roger Howe in Yale's Mathematics Department with the simplified statement of the problem where we let x stand for vx2, and use only two rather than three dimensions. He easily showed me that Maxwell's conclusion is in fact obvious.
set y = 0
2) f(x) * f(0) = g(x)
or, substituting x+z for x
3) f(x+z) * f(0) = g(x+z)
equating the expressions for g(x+z) in 1 and 3
4) f(x) * f(z) = f(x+z) * f(0)
divide by f(0)2
and define F(x) = f(x) / f(0)
5) F(x) * F(z) = F(x+z)
That is, we do not need to deal with separate f and g functions. Each is just a constant [f(0) for f, f(0)2 for g] times F. From here on in we need consider only F. [It will turn out below that f(0) = f(0)2 = 1]
The equation in F is well studied and known to have exponentials as its only solutions. This would almost certainly have been well known to Maxwell in the 1850s.
F(x) * F(1) = F(x+1)
i.e. adding one to the argument multiplies the function by the constant F(1).
Fill in the function by going to half-integral intervals
F(x) * F(1/2) = F(x + 1/2)
thus F(1/2)2 = F(1)
We can do this with any rational fraction,
Then fill in the remainder by continuity.
This is in fact how the exponential is constructed, if you do it carefully.
And of course it turns out that f(0) is 1