# Understanding a numerical integration method

I am working on a project that uses numerical integration to solve some differential equations. Not coming from a solid math background, I have a problem understanding some integration methods. Specifically, I have the following equation:

$$\frac{d y}{d t} = \frac{u - y}{v} \,,$$ where $u$ and $v$ are constants and $y(0)$ is known.

I solved this initially with a simple Euler integration and then with some other higher-order methods (Runge-Kutta 4). Then, I found another numerical integrator that did: $$y_{n+1} = y_{n} + (u - y_{n})(1 - e^{\frac{-\Delta t}{v}}) \,.$$

My question is: what is this integration method? I checked and the results are basically the same, but I had no idea about this technique and I would like to know more about it.

## 1 Answer

Its just the analytical answer to your equation.

$y(t) = c_1 e^{-\frac{t}{v}}+u$, with an initial condition $y(0) = c_1 + u = y_0$ which gives you $y(t) = (y_0-u) e^{-\frac t v}+u = y_0 + (u-y_0)(1-e^{-\frac t v})$

Just call $y_{n+1} = y(t)$ and $y_n = y_0$ with $\Delta t = t$ and you obtain your equation. This is not a general numerical method which applies to other differential equations.