September 11, pt. 2

Approaching Equilibrium: Introduction to Solving Differential Equations (Section 2.4.1)

This lecture introduces the fundamental concept of approaching equilibrium by exploring how to solve a specific type of equation, laying the groundwork for more detailed discussions on Thursday.

The Differential Equation Under Consideration

We are looking at an equation of the form:
$\frac{dx}{dt} = f(x)$

• $\frac{dx}{dt}$: This term represents the rate of change of a variable $x$ with respect to time $t$. It describes how $x$ is changing moment by moment.

• $f(x)$: This is a function that dictates the rate of change. The rate at which $x$ changes depends on the current value of $x$ itself.

The Problem: Predicting Future States Given Initial Conditions

Suppose we are given an initial condition:

• At a specific initial time, say $t<em>0 = 0$, we know the exact value of $x$. Let's call this position $x</em>0 = x(0)$. (E.g., at time zero, we are at position zero, $x(0)=0$).

The Core Question: Where will $x$ be at a future point in time, specifically at $t = t_0 + \Delta t$?

• We want to know $x(t_0 + \Delta t)$, where $\Delta t$ is a very small, infinitesimally small time increment, not far from the current moment.

Method of Approximation: Using the Rate of Change (Implicit Euler's Method)

Instead of direct integration (a more advanced technique), we can approximate the future state using the known rate of change.

1. Understanding $\frac{dx}{dt}$ Concretely: If $x$ represents your position, then $\frac{dx}{dt}$ is your velocity. Velocity has units of position per unit time (e.g., meters per second, or units of $x$ per units of $t$).

2. Calculating the Change in $x$ ($\Delta x$):

   • At the initial time $t<em>0$ and position $x</em>0$, we know the rate of change is given by $f(x_0)$. This is essentially the