follow up to previous question " is this an impulse response or step response"
Let me provide an answer,

1-) In the answer I have used the notation $x(t)=\delta(t)$ and $x(t)=u(t)$ to indicate that the input to the system is a dirac impulse and unit-step function respectively. The correct notation should be $x(t) \equiv  \delta(t)$ which means identical, by definition. Triple equality sign is almost always omitted and the double equality sign is used instead in signal processing context. (it's therefore not asking for the solution of an equation, which is generally understood from double equality sign).

2-) The subject of differential and difference equations is rich. According to the usual literature, continuous time-systems are described by differential equations and discrete-time systems are described by difference equations. When a continuous-time signal is sampled and the associated continuous system is transformed into a discrete-time system the corresponding differental equation is also discretized and converted into an equivalent difference equation. One application of such discretization is the usual Kalman filering equations.

I've analysed the continuous-time leaky integrator model whose input-output relationship was assumed by definition as $y(t) = \mathcal{T} \{x(t) \} = \int_{-\infty}^{t} e^{-\alpha(t-\tau)} x(\tau) d\tau $. Given this definition of the system the corresponding continuous-time LCCODE is obtained as in the answer. In the web-link only the differential equation relation is provided as $ dy/dt = s - y(t)/k $ where $y(t)$ is the water level (the output) $s=s(t)$ is the input and $1/k$ is the leakage constant, the differential equation is then $$ \frac{dy}{dt} + \frac{1}{k} y = s $$ or equivalently $$ \frac{dy(t)}{dt} + \frac{1}{k} y(t) = s(t) $$ This differential equation is the same as $ y' + \alpha y = x$ after replacing $\alpha = 1/k$ and $s(t) \equiv x(t)$.

On the other hand in your original question, you refer to a discrete differential equation that I cannot clearly understant whether it's a difference equation or a differential equation (the terms discrete and differential used together as a mix) or it's just another application of discretization as I defined above.

However the following excerpt from the original web-reference indicates to me that:


  We can easily simulate this leaky integrator in discrete steps of time
  by changing the value of 'y' on each step according to the equation
  above:


There is only a continuous-time differential equation and it's solution is numerically computed by the software, which is indicated as the simulation of the ODE using discrete steppings (the numerical computation)

So as I understood given the continuous time differential equation
$$ y'(t) + \alpha y(t) = x(t) $$ one can reach the numerical version as
$$ \frac{d y}{dt} = -\alpha y(t) + x(t) $$
$$ dy = ( -\alpha y_t + x_t) dt $$ 

Where $y(t)$ is replaced by $y_t$ just as a matter of notation. So they are the same thing. The web-link numerically solves the contiuous differential equation for various input stimulus.