I was toying with equations of the type f(x+α)=f′(x) where f is a real function. For example if α=π2 then the solutions include the function fλ,μ(x)=λcos⁡(x+μ). Are there more solutions?On the other hand, if I want to solve the equation for any α, I can assume a solution of the form f(x)=eλx, and find λ as a complex number that enables me to solve the equation...I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator ϕ:f(x)→f(x+1) adds another one? Is there some litterature about this kind of equations?

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Melissa Moore · Accepted Answer

This sort of thing is known as a delay differential equation, and there are lots and lots of books on this subject. They&#039;re called &#039;&#039;delay&#039;&#039; because the constant α is usually chosen to be negative, so that f′(x) depends on the values of f at earlier times.The equation you have given has an infinite dimensional space of solutions. In particular, if g|0,g∣→R is any smooth function satisfying g(n)(α)=g(n+1)(0) for all n≥0, then there exists a unique extension of g to a smooth solution f defined on the entire real line. For x≥0, this solution is defined byf(x)=g(n)(x−nα) where n=[xα]while for x≤0 it involves iterated integrals of g.

Esther Phillips · Accepted Answer

Integers without large

alenahelenash · Accepted Answer

Note that f(x)=eλx is solution of f′(x)=f(x+α) if λ=eλα, and this equation is satisfied by infinitely many complex λ if α≠0, namely λ=−wn/α where wn are the branches of Lambert W(−α)

I was toying with equations of the type f(x+\alpha)=f'(x) where f is a real function. For exam

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