I'm trying to solve the following:
(1)$\ {d\over dx}[f(x)] = x^2*f(x-1)$
Motivation: The factorial (or gamma function in discrete domain) obeys the following two equations:
$f(x) = x*f(x-1)$
$\Delta_{x}[f(x)] = x*f(x)$
Combined:
(2) $\Delta_{x}[f(x)] = x^2*f(x-1)$
Which makes (1) the continuous analog of (2). (2) has a solution:
$f(x) = x!*(c_{2}((-1)^{x+1}\Phi(-1,1,x+1)+\ln(2)) + c_{1})$
$=x!$ for $c_{1}=1$ and $c_{2}=0$
My initial approach for (1) was to write it as an integral and, taking advantage of integration by parts, iterating it many times:
(3a) $f(x) = \lim_{n\to \infty} [\sum_{k=1}^n [P_{k}(x)*f(x-k)]]$
Where:
(3b) $P_{k}(x) = \sum_{j=k+2}^{3k} [(-1)^{j-1}*C_{j}*x^{j}]$
I created a Matlab script to store the $C_{j}$, which are fractions. But the numerators/denominators don't seem to have a closed-form expression, even checking OEIS.
Am I on the right track or is there a better approach? Thanks
Edit: I'm sorry, why on earth is my question downvoted? Is the question just difficult? I've explained my progress. I don't understand.
(1) is actually a linear equation. This is a first order delay differential equation with one discrete delay. Try the Method of Steps to get a recursive solution. An equation like this requires initial data from an interval of length 1 (since the delay is 1), i.e. $f(x) = \phi(x)$ given for $x\in[\xi-1,\xi]$ for some $\xi\in\mathbb{R}$ (usually $\xi = 0$). The Method of Steps just rewrites the DDE as an integral, and integrates sections of length 1 starting from the initial data.
Edit: I'm not sure why people are downvoting your question, but yes this question is quite difficult. DDE's are an active area of research, and we don't have much in the way of analytic closed-form solution for these equations. For constant coefficient equations, we can use the Method of Characteristics, but with variable coefficients, the Method of Steps is the best we have in general. There may exist some specific techniques for dealing with simpler equations, but I'm not aware of them.