My boyfriend is a huge nerd and likes to talk about physics. Most of the things he tells me I can understand but I still can’t grasp differential equations. I love talking about physics and programming with him and I just wanna understand. Thanks!
In: Physics
When the rate of change is dependent on the current value and not independent of it, differential equations are the lever to reach for. Think population growth, radioactive decay, etc.
Sometimes it is very hard to look at a scenario and come up with a math equation that can tell you what that scenario is going to do. However, a lot of times it’s very easy to look at a scenario and come up with a math equation that describes what it is doing at that exact moment.
Lets use rabbits as an example. Imagine I tell you that I have five pairs of rabbits. Every month each pair of rabbits will produce five more pairs of rabbits.
This is a pretty simple scenario. If you’ve taken some algebra classes you might be able to recognize this as an exponential growth function, but lets just assume you have no idea how to come up with a math equation to describe how many rabbits there are at any given time. Just reading the scenario and coming up with an equation would be pretty difficult. Instead try and come up with an equation that explains how fast the rabbits are growing. That’s actually pretty easy.
r’ = 5*r
Where r’ is how fast the rabbits are growing per month, and r is the number of pairs of rabbits. This function has both rabbits (r) and the growth rate of rabbits (r’). We call an equation that contains the rate of a variable and that variable a differential equation.
Once we have a differential equation we can do a bunch of mathy things to it to turn it into a normal equation. The rabbit scenario was pretty simple, but there can be scenario’s where it can be almost impossible to come up with an equation without starting at a differential equation.
You know when you pour something out of a bucket, and it starts off fast but gets slower and slower as there is less and less water? Thats a simple form of a differential equation.
The output of the thing changes based on how the thing has been going. Fires burn hotter when there is more fuel, but use up their fuel so slow down is another example. Differential equations are just math where the answer depends on itself a bit.
They are equations that relate functions to their derivatives. Which isn’t much help unless you have an idea what that means already…. but basically they are equations which consider the rates of change between different variables and how to get a meaningful answer out of a system with several variables if you have these rates of change for the variables as they relate to one another.
So if you can get past my mealy-mouthed explanation, the takeaway here is that they are an incredibly powerful way of describing (sometimes approximately, especially when you get to more complex systems) how much stuff will do what, at what sort of rate.
This is vital to physics, but also has extremely wide applications in pretty much all the sciences.
In regular algebra, we relate variables directly to each other. y=2x means that for every x there’s 2 y’s.
In physics, you often want to talk about how things change in relation to each other, or can only write down an equation talking about how things change. That’s where differential equations come in.
One way of writing something like this is to use a “d” with the variable, or a funny curly-looking “d”. So if you see “dx” it means a tiny little change in x. Mathematically speaking it’s an infinitesimally small change in x, but thinking about it as just “really really small” works for most purposes.
When we do this we can start writing equations like dy=2dx. This is a way of saying that if we change x by a teeny tiny amount, the amount that y changes by is twice as much. If we solve this we get the equation y=2x+C where C is some arbitrary constant, and as you can hopefully see (remember y=mx+b?), it’s a line with a slope of 2. It’s a line whose slope goes up twice as fast as it goes to the right, which is what we said when we started.
In physics this pops up *all the time*. We want to talk about how heat changes when we pump energy into a system, or how the speed of an object changes as we apply force, or how the resistance of a circuit changes as the current changes…the list is huge.
Regular equations talk about how some value changes as something else changes. If you’re driving a car, your speed will vary depending on your acceleration, and your distance will vary depending on your speed.
But lots of things are more complicated than that. Many things will change depending on a difference between things. For example, how quickly a mug of coffee cools depends on the difference between the temperature of the coffee and the temperature of the room. At first, this difference is big, meaning the coffee cools very quickly. But as it gets cooler, the difference gets smaller. This means it will cool more slowly. So maybe in the first 10mins it drops by 20 degrees, but in the next 10mins it only goes down by 5 degrees. This is what differential equations do. Instead of something being related to the value of something else, it’s related to how something else changes (over time or over space).
The one I’ve described is called the heat equation. Another common differential equation is the wave equation. Think of a spring. The force it pulls with is directly related to how stretched it is. If you pull it twice as far, it pulls back twice as hard. Force is related to acceleration. This means the acceleration is directly related to the distance you pull it. This results in something called sinusoidal motion, the classic boing motion of a spring. But it comes up everywhere!
Surely you know about recurrence relation right?
Here is a simple example:
I am walking. Each second, I walk 2 steps. After 60 seconds, I will walk 120 steps.
“Each second, I walk 2 steps” is a simple example of a recurrence relation. It tells you the relationship between the number of steps between one point in time that 1 second afterward. Once you know the recurrence relation, you can attempt to make long-term prediction, such as “After 60 seconds, I will walk 120 steps”.
Recurrence relation is important, because a lot of time, short-term relationship is easy to describe, but long-term relationship are not. “Every year I gain 10% in compound interest” is something you might heard, but you have to put in effort to figure out how much money you will gain after 10 years.
Now, differential equation is very similar to recurrence relation. The difference between them is that recurrence relation can describe how things change between discrete step in time (like “after 1 day”), but differential equation is needed to describe how things change continuously over time. This introduces some additional difficulty, because there are no “next” step, so differential equation only describe how things change over an arbitrarily short interval. For the purpose of simulation, people often treat differential equation as recurrence relation anyway (by picking a very short time interval and pretend that the equation is talking about discrete time step).
Why is differential equation important in physics? Because time is believed to be continuous, and physical law are described in term of relationship between short-term changes in physical quantity. For example, Newton’s law of gravitation describe how short-term change of objects’ positions are related to objects’ mass and distance. And the reason why we focus on short-term relationship is because these are the one that we can experiment on and understand intuitively.
Once you have a differential equation, you can attempt to solve them. It’s usually very hard, however.
There *are* physical laws that does not rely on short-term changes, but instead focus on long-term relationship between physical quantity. But generally they are derived *after* we have our differential equation, not before. Also, they are less intuitive to understand. For example, Lagrange’s mechanics posit that an object will choose a path that minimize its action. This is not intuitive to us, because it seems to say that an object has a will of its own and can see the future. But Newton mechanics is intuitive: things push on each other, inflict force, and force change velocity. But this require differential equation.
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