Chill out, don’t worry, because I’m here to break it down in the most casual way possible.

To kick things off, let’s define what we mean by “approximating” when it comes to solving these ***** little beasts. Essentially, we want to find a value that is close enough to the actual solution without having to go through all the hassle of finding an exact answer. Because who has time for that?

Now, there are many different methods for approximating solutions to differential equations, but today we’re going to focus on one in particular Euler’s method. This is a simple and straightforward approach that involves breaking down the equation into smaller chunks and solving each chunk individually. It may not be the most accurate or efficient way of doing things, but it’s definitely the easiest to understand for those of us who struggle with math.

So how does Euler’s method work? Well, let’s say we have a differential equation that looks like this:

y'(t) = f(t, y(t))

This basically means that if you know the value of y at some point in time (let’s call it t0), and you also know how fast y is changing over time (which we can calculate using the derivative), then you should be able to figure out what y will be at a later time (t1).

But instead of trying to solve this equation for every single point in between, Euler’s method suggests that we break it down into smaller chunks and approximate each chunk individually. So let’s say we want to find the value of y at t1, which is some distance away from our starting point (t0).

First, we calculate the slope of the line that connects the points (y(t0), f(t0, y(t0))) and (t0+h, f(t0+h, y(t0))):

m = [f(t0+h, y(t0)) f(t0, y(t0))] / h

This gives us the slope of our line segment. Next, we use this slope to find a new point on the line that is some distance away from (y(t0), f(t0, y(t0))) in the direction of increasing x:

x1 = t0 + h
y1 = y(t0) + m * h

This gives us an approximation for what y will be at time t1. And that’s it! We can repeat this process over and over again to approximate solutions to more complex differential equations, as long as we keep breaking them down into smaller chunks.

It may not be the most accurate or efficient way of doing things, but it’s definitely the easiest to understand for those of us who struggle with math.