But don’t freak out, because we’re going to break it down in the most casual way possible.

First things first, what are these ***** beasts? Well, they’re just fancy math problems that involve taking derivatives of a function more than once (hence “higher-order”). And if you thought calculus was hard enough with just one derivative, wait until you see the chaos that ensues when we start adding more.

But don’t freak out, because there’s a way to make these problems less painful by approximating their solutions using numerical methods. That’s right, Instead of trying to solve them exactly (which is basically impossible for anything beyond a first-order equation), we can use computers to find an approximation that’s good enough for our purposes.

Now, there are many different ways to do this, but one popular method involves using something called “finite difference methods.” Essentially, what you do is break up the interval where your function lives into smaller and smaller pieces (called “steps”), and then use a formula to approximate each derivative at those points.

For example, let’s say we have a second-order differential equation:

y”(x) = x^2 + sin(x)

We want to find an approximation for y(1), which is the value of our function when x=1. To do this using finite differences, we first choose some small step size (let’s say h=0.01). Then, we calculate the values of y and its derivative at each point in the interval [0, 1] using:

y(x) = y(x-h) + hy'(x-h)
y'(x) = y'(x-h) + h y”(x-h)

where y’ is the first derivative of our function. We can then use these approximations to calculate an approximation for y(1):

y(1) y(0.99) + hy'(0.99)
= y(0.98) + h[y'(0.98) + hy”(0.98)]
= … (repeating this process for each step until we reach x=1)…

We have an approximation for our solution that’s good enough for most practical purposes.

Of course, there are many other numerical methods out there besides finite differences some of which are more accurate and efficient than others. But the point is this: we don’t need to solve these ***** higher-order differential equations exactly in order to get useful results. We can use computers to find an approximation that’s good enough for our purposes, and save ourselves a whole lot of headaches (and math anxiety) along the way.

Now go out there and impress your friends with your newfound knowledge!