I want to mathematically find a function that is
1. continuous
2. approach 0 at +inf and -inf
3. does NOT approach inf at 0
4. symmetrical
5. always positive
basically i know that the answer is e^(-x^2)
which is the normal distribution equation,
but i want to find it myself,
to build a set of differential equations that upon solving will give this result
my passion for this is big,
i have been thinking about this for days
but all i got so far is
f(x) = f(-x)
df(x)/dx = -df(-x)/dx
any ideas how to approach this beast?
2026-03-25 18:14:12.1774462452
What is the right approach to find a function?
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Your conditions are not sufficient to find out a particular family of solutions by a set of equations for example also $f(x)$ such that
satisfies the given conditions.