Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below: $$f'(x)=f(x+c),$$ where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.
I found two families of functions (i.e. exponential functions and sine/cosine functions) that satisfy this differential equation. For example, $f(x)=c_1 \sin(x+c_2)$ and $f(x)=c_1 e^x$.
Can you come up with any more function families or prove no more functions exists?
Following Euler, let us look at solutions of the form $f(z)=e^{sz}$, where $s$ is a complex number. We obtain a transcendental equation $$s=e^{cs},$$ which for every complex $c\neq 0$ has infinitely many complex solutions $s_n$. Now any linear combination $$\sum_{n}a_ne^{s_nz}$$ with complex $a_n$, and any limit of such linear combinations will satisfy your equation. This gives all solutions.
The method actually applies to a wide class of linear equations of "convolution type". See Malgrange's approximation theorem (Theorem 16.4.1 in Hormander, Analysis of linear partial differential operators) and also these entries on MO: Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$, On equation $f(z+1)-f(z)=f'(z)$
There is a well-established method to obtain solutions to this equation, indeed all solutions. It is called the Fourier transform. Its application leads to an algebraic functional equation which can easily be solved to give the solutions already deposited here.
Fair warning. The functional equation seems at first sight to have only the trivial (i.e., zero) solution—it must be dealt with in the distributional sense. The resulting solutions are suitable combinations of delta distributions located at the (complex) zeros of the function displayed in the solution of Alexandre Eremenko—with the exception of the double one which requires, in addition, a derivative thereof.
Warning no. 2. The F.T. can be used for ALL distributions, say on the line, and the techniques used here (for example, dividing by smooth functions) are very standard and simple—they have been used and loved for about 60 years. The transform of such a distribution is a so-called analytic functional. (Recall that a distribution is defined as an element of the dual of a suitable l.c.s. of test functions on the line. An analytic functional is the dual of a suitable l.c.s. of entire funcions. Prominent examples of such functionals are point evaluations at complex numbers, often called delta functions for obvious reasons. Note that restriction to the real line injects the space of entire functions into a space of smooth functions on the line and this shows how classical distributions on the line induce in a natural way analytic functionals, thus displaying them as genuine generalisations).
The basis for this use of the F.T. for all distributions is the Paley-Wiener-Schwartz theorem which establishes the fact that it is an l.c.s. isomorphims between the standard space $\cal D$ of test functions and a suitable space of entire functions defined by growth conditions. This can then be dualised to establish one between $\cal D’$ and a space of analytic functionals.
Useful references. The classical monographs of Schwartz and Stricharzt.
