What is Gamma density's differential equation? What is Gamma density's differential equation?
 A: Well, I agree with the comment that there is no one differential equation whose solution is the gamma-distribution, here is an attempt to give an example how such equation can appear. 
Consider the so-called frailty model [1]: it is assumed that the hazard rate of an
individual is given as the product of an individual specific quantity $Z$ and a basic rate $\alpha(t)$:
$$
\alpha(t|Z) = \alpha(t)Z \tag{1}
$$
Given $Z$ the probability of surviving up to time $t$ is
$$S(t|Z) = e^{−ZA(t)},\, A(t) =\int_0^t \alpha(\tau)\,d\tau.$$
The population survival function is therefore
$$
S(t) = \mathbf{P}(T > t) = \mathbf{E}(e^{−ZA(t)}) = \mathbf{M}(−A(t)),
$$
where $\mathbf{M}(\lambda)$ is the moment generating function (mgf) of the distribution of $Z$. The distribution of the quantity $Z$ will be changing with time, but an interesting fact is that the moment generating function at any time moment can be expressed through the mgf at the initial time:
$$
\mathbf{M}(t,\lambda)=\frac{\mathbf{M}(\lambda-A(t))}{\mathbf{M}(-A(t))}.\tag{2}
$$ 
Now assume that $A(t)=-t$. The big question is how the initial distribution will be changing with time. However, it is possible to look from the other side: fix some conditions on how the distribution changes and ask which initial distribution satisfies these conditions. For example, from $(2)$ we have that
$$
\frac{\mathbf{M}''(t)}{\mathbf{M}(t)}-\left(\frac{\mathbf{M}'(t)}{\mathbf{M}(t)}\right)^2=\sigma^2(t),\tag{3}
$$ 
where $\sigma^2(t)$ is the variance of the distribution of $Z$ at time $t$. Let's assume that $\sigma(t)=\sigma$ and does not change with time, then we get a solution to $(3)$ which is mgf for the normal distribution.
Finally, one of the answers for your question. If we assume that the coefficient of variation is constant with time, then the differential equation $(3)$ has the solution which is mgf for the gamma distribution. More precise, the coefficient of variation is 
$$
CV=\frac{\sigma(t)}{\mathbf{E}(Z(t))}, 
$$
and the equations becomes, with $m=m(t)=\mathbf{M}(t)$ 
$$
\frac{\ddot m}{m}-\left(\frac{\dot m}{m}\right)^2=CV^2\left( \frac{\dot m}{m}\right)^2,
$$
with the initial conditions $m(0)=1,\,m'(0)=\mathbf{E}(Z(0))$.
[1]: O. O. Aalen, Ø. Borgan, and H. K. Gjessing. Survival and event history analysis: a process point of view. Springer Verlag, 2008.
A: Methinks the previous answers were to a question other than the one asked. For the gamma distribution's Probability Density Function -- 
f(x) = Gamma(x,α,β) = [β^α x^(α-1) e^(-βx)] / Γ(α), or alternatively with parameters k,theta -- 
there can be only one solution to the differential, f'(x)=dy/dx. The function has only one slope at any given x. 
