2
votes
2answers
124 views

Unusual Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
4
votes
1answer
303 views

A Differential Equation with Nested Functions

This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems. I would like to know whether there exists a differentiable function from the ...
2
votes
0answers
137 views

Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation $$ \text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t) $$ with ...
0
votes
2answers
525 views

Approach to solving a differential-functional equation

What could be an approach to solving such equations? $$f'(x)=C \prod_{k=0}^x f(k)$$ $$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$ Here the product and the sum are understood as indefinite sum and ...
1
vote
1answer
297 views

Does any iterative equation of n-th order have exactly n independent solutions?

Does any iterative equation of n-th order which does not inclute derivatives of order higher than 1 have exactly n independent solutions? Let's designate n-th iterate of a function $y(x)$ as ...
12
votes
2answers
918 views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
2
votes
1answer
382 views

What kind of uniqueness can I conclude for solutions to a simple functional equation?

I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness. At the most vague version, I am in ...
11
votes
1answer
582 views

Modular equations for quasimodular forms

This problem is motivated by this question and by teaching modular polynomials for the classical modular invariant $j(\tau)$. The latter implies that if we consider the fields of modular functions ...
5
votes
1answer
703 views

beyond differentially algebraic power series

In a recent question, we learned about the existence of functions that do not satisfy any algebraic differential equation. One nice property of such equations is that there is a good way to ...