So, I am working on a game, and I need to come up with some math formulas for a proof of concept (and tweak them to make sure it's balanced). In particular, I have a quantity $y$ that should decrease over time with a specific pattern. As is often the case in games, this quantity will be stored in a variable, and the new value of this variable will be re-computed every frame starting from the current value and the time passed since last frame $\Delta t$.
Through trial and error, I came up with this formula to compute the new value of $y$: $$ y_{new}=\frac{y_{old}}{((k)^{y_{old}})^{\Delta t}} $$ where $k$ is a constant parameter such that $k>1$, $y$ will always be $>1$ (meaning that if it goes below 1 I will discard the value and consider $y=1$ instead), and $\Delta t$ is a value in seconds always $>0$. Now, this "formula" is okay from a coding perspective, it's ready to be implemented, but I would like to be able to plot it on Geogebra or Wolfram Alpha in order to analyze and tweak it as needed, and for that I need to transform it in a "traditional" function $y=f(t, y_0)$. How do I do that?
I don't even know what to Google (I guess it's something in the realm of differential equations?), so it would be nice if you could explain the reasoning/point me to a general explanation so I can further research the topic, as I expect that I will face similar problems multiple times over the course of the development.
What you are looking to solve for your function is, as you correctly noted, a differential equation. (Or rather, a difference equation since you are taking discrete steps of time, rather than continuous ones, but the two are intimately related, in any case.)
Your proposed differential equation gives a nice, simple solution provided the time steps are small. If we take logarithms of both sides, we can convert the multiplicative operations into additive ones
$$y(t_0+\Delta t)=\frac{y(t_0)}{k^{y(t_{0}){\Delta t}}} \\\ln y(t_0+\Delta t)=\ln{y(t_0)}-{y(t_{0}){\Delta t}}\ln\left({k}\right)$$
Denoting $\ln y(\cdot)$ by $L(\cdot)$ and rearranging, we have
$$\frac{L(t_0+\Delta t)-L(t_0)}{\Delta t}=-\ln k \, e^{L(t_0)}$$
where the LHS is a difference quotient. Provided $\Delta t\approx 0$, we can consider this to be approximately the derivative of $L$. I'll leave it to you to solve $L'(t)=-\ln k \, e^{L(t)}$ and back-substitute for $y$.