You've written down the equation that describes the rate of your ruin. Now, the real work begins: finding the function that tells you exactly how your misery unfolds over time. This is called solving a differential equation. It's like having a recipe for how fast something changes and using it to figure out the full story of its growth, decay, or oscillation.
Solving differential equations can be incredibly complex, requiring advanced mathematical techniques. But for simple cases, the idea is to find a function whose derivative matches the relationship described in the equation. If $\frac{dy}{dt} = f(t, y)$, we are looking for a function $y(t)$ such that when you take its derivative with respect to $t$, you get $f(t, y(t))$.
Finding these solution functions is what allows algorithms to predict your trajectory. By solving the differential equations that model your behavior, they can estimate your future state – how much time you'll spend online, what content you're likely to engage with, or when your motivation will hit rock bottom.
Let's look at the solutions to the simple differential equations we set up in the previous module. Observe how the equation describing the rate of change leads to a function describing the quantity itself over time.
Given a differential equation relating $\frac{dy}{dt}$ to $t$ and $y$, the goal is to find the function $y(t)$ that satisfies this equation. This function $y(t)$ is called the general solution.
For example, if the equation is $\frac{dy}{dt} = 2t$, we are looking for a function $y(t)$ whose derivative is $2t$.
Solving differential equations involves various techniques depending on the type of equation. For simple equations, methods like integration or separation of variables can be used.
Consider the equation $\frac{dy}{dt} = 2t$. We can integrate both sides with respect to $t$:
$\int \frac{dy}{dt} dt = \int 2t dt$
$y(t) = t^2 + C$
Here, $C$ is the constant of integration. This is the general solution.
Consider the exponential growth equation $\frac{dA}{dt} = kA$. This can be solved using separation of variables:
$\frac{dA}{A} = k dt$
$\int \frac{dA}{A} = \int k dt$
$\ln|A| = kt + C'$
$|A| = e^{kt + C'} = e^{C'} e^{kt}$
$A(t) = Ce^{kt}$ (where $C = \pm e^{C'}$ or 0)
This is the general solution for exponential growth.
The general solution $y(t)$ describes the family of all possible functions that satisfy the differential equation. It includes an arbitrary constant (like $C$).
In the visualization, we show a simple differential equation and its general solution. The equation describes the rule of change, and the solution describes the outcome of that change over time.
Understanding how to solve these equations is key to predicting the behavior of dynamic systems, from the decay of your motivation to the growth of algorithmic control.