Is the Arrhenius Equation a solution to a differential equation?
To answer the question of the title. In Adam-Gibbs theory the activation enegy is defined as [1]:

$$E_\mathrm a=-R\left[ \frac{\partial \ln k}{\partial (1/T)} \right]$$
Now clearly this is a first order linear differential equation, as you were looking for. This can easily be solved by seperation of variable:
$$\int E_\mathrm a\, \mathrm d(1/T) = -R\int \mathrm d\ln k $$
$$\frac{E_\mathrm a}{T}=-R\ln k+A'$$
$$k(T)=A\exp\left( \frac{-E_\mathrm a}{RT} \right)$$
Here $A=\exp(-A'/R)$, now it can be seen that the Arrhenius equation has been regained.

[1] S. Petrucci; Ionic Interactions: From Dilute Solution to Fused Salts; equation 125