You understand the rate of change – how fast your motivation is plummeting or your anxiety is skyrocketing. Now, let's capture that dynamic mathematically. A differential equation is an equation that relates a function to its derivatives. It's a way of saying, "The rate at which this quantity changes depends on the quantity itself, or on time, or on other factors."
Think about modeling the growth of your unread email inbox. The rate at which emails accumulate ($\frac{dI}{dt}$) might be constant (people send you emails at a steady rate) or it might depend on how many emails you already have (the more emails you have, the more overwhelmed you feel, and the less likely you are to delete any, effectively increasing the net rate of accumulation). Setting up the differential equation is the process of translating these real-world (or your internal, despair-fueled) relationships into mathematical form.
This is where the power of calculus becomes clear. Instead of just observing the rate of change at a single point, we write an equation that describes the rate of change everywhere. This equation is the key to understanding the entire trajectory of the system, not just a snapshot.
Let's look at how to set up a few simple differential equations. The visualization below conceptually represents these mathematical models.
First, identify the quantity you want to model that changes over time. Let's call it $y$. This could be your anxiety level, the number of unread emails, or the amount of work remaining. Its rate of change with respect to time $t$ is its derivative, $\frac{dy}{dt}$.
Example: Let $A(t)$ be your level of anxiety at time $t$. Its rate of change is $\frac{dA}{dt}$.
This is the core of setting up the differential equation. How does the rate of change ($\frac{dy}{dt}$) depend on the quantity itself ($y$), on time ($t$), or on other constants or variables?
Simple Example 1: Constant Rate of Change
If your unread email inbox grows by a constant rate (e.g., 10 emails per day), the differential equation is:
$\frac{dI}{dt} = 10$
Here, the rate of change of the number of emails ($I$) is simply a constant.
Simple Example 2: Rate Proportional to the Quantity
If your anxiety level ($A$) increases at a rate proportional to your current anxiety level (the more anxious you are, the faster you get more anxious – a common feedback loop!), the differential equation is:
$\frac{dA}{dt} = kA$
Here, $k$ is a positive constant of proportionality. This is a classic model for exponential growth (or decay if $k$ is negative).
Simple Example 3: Rate Depends on Time
If the rate at which you complete work decreases linearly as a deadline approaches (time $t$), the differential equation might look like:
$\frac{dW}{dt} = -c t + b$
Where $W$ is the amount of work remaining, $c$ is a positive constant (rate of decrease of work completion rate), and $b$ is another constant. The negative sign indicates that the amount of work remaining is decreasing (you are completing work, albeit at a decreasing rate).
Once you have expressed the relationship between the quantity and its rate of change mathematically, you have set up the differential equation. This equation is a concise summary of the dynamic process you are modeling.
The visualization area above is conceptually where these equations reside, representing the mathematical models of your various forms of misery. The next step is to solve these equations to find the actual functions that describe the trajectory over time.