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We’re taking very basic differential equations in school and I know how to solve them (more or less) but I think I’d be better when I u derstand what this is doing. Like we use differentiation to find the slope of a curve integration to find the curve equation from a given slope of a point or find the area under the curve

When I solve a differential equation. What is it that I do?

In: Mathematics

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Well, I like to remember that Newton invented calculus to explain things in motion. So a practical example often helps.

Like we know distance as here to there.

And speed is d = r/t

And Acceleration is v = r/t/t

Those are differential moments. If we know your position, given by some function, then your speed can be derived from distance from the initial point as a function of time. and your rate of speed over time tells use your positional Acceleration.

So what the differential calculus is helping us to understand is that if your position changes we can figure it out where you are by the rate of change over time. And even more accurately if we know the rate of change at a specific time (or as your rate of velocity changes over time)

And, in a concrete example. If you throw a ball, and it’s arc in space is described by some function. We can calculate its distance, velocity and speed at different points in time, using differential calculations (the ball goes up, but is acted on by wind and gravity) so it’s not a straight line – but a function that we can manipulate

Differential equations are different from differentiation and integration, but they’re related.

Imagine you have a boiling pot of water that you lay out on the table. You’re curious about how fast it will cool down. Your natural intuition tells you that it’s cooling down faster when the water is still really hot, but cools slower as the water gets closer to room temperature.

You would be right this is called Newton’s law of cooling. It states that the rate of cooling of an object is proportional to how much difference there is between the object and the environment. So let’s write out that equation. T’ will be the rate of change of temperature, A will be the temperature of the room, and T will be temperature of the pot of water.

T’=(A-T)

Seems supper easy right, but take a closer look. That equation has the speed temperature is changing, and also temperature. That’s a differential equation. When your rate of change is determined by the thing that’s changing.

Now what would be really useful is if we could somehow figure out a formula based on time from that. You can do it, and that’s what you’ll learn in a differential equations class.

You are looking at this wrong.

Dif Eq is about how to use and solve for differentials in algebraic-like equations. You learn to solve the dif eq for any component of the dif eq.

Dif Eqs are used extensuvely to model complex physical systems. A good example…a car is driving down a hill. The driver pushes the gas pedal to the floor. What happens?

Note that the question is very open ended and general. This is a real question for the real world.

To answer the question, you have to build a dif eq.

The components:

1. The velocity, which is derivative of position.

2. The acceleration of the car based on gravity.

3. The acceleration of the car based on the engine power and the weight of the car. The weight of the car depends, in part, on how fast the fuel is consumed by the car.

4. The wind resistance, which is based on the velocity of the car and the air density.

5. The air density is based on the altitude of the car, which is based on the slope of the hill and the velocity of the car.

And on and on and on…you can go further down the rabbit hole.

Let’s imagine a big empty field, surrounded by walls. We plant some grass seed all over the field. Each day, the grass will grow at some rate. The rate isn’t constant, or the grass would eventually be so tall that it would be up in space. So the rate of change of the amount of grass is dependent on how much grass there is already.

Now let’s put some mice in the field. The mice eat grass, and if they have enough grass to eat, they will live long enough to reproduce and make more mice. At first, with a small number of mice and plenty of grass, the mice will reproduce a lot and there will still be a lot of grass. But as time goes on, eventually the rate that the grass grows each day won’t be enough to keep up with the number of new mice. Some of the mice will starve to death and not reproduce.

So now for the grass, its rate of change each day depends on how much grass there is already and how many mice are eating it.

And for the mice, the change in their population each day depends both on how many mice there are (more reproduction) and how much grass there is (less reproduction).

Finally, let’s add some cats. The cats only eat mice, and they need quite a lot of mice in order to live long enough to reproduce. Just like the mice with the grass, the cats will reproduce more if there are lots of mice, and less when there aren’t enough mice.

It shouldn’t come as a surprise that we now have a couple more relationships: the change in the number of mice now depends on how many cats there are (cats eat mice), and the number of new cats depends on how many mice there are (cats can die from starvation before they can reproduce).

All of these relationships involve both *amounts* and *rates of change*. Let’s look at all of the dependencies we’ve found:

Grass rate of change: how much grass, how many mice

Mice rate of change: how much grass, how many mice, how many cats

Cats rate of change: how many cats, how many mice

When we write out all of these relationships with specific mathematical values, we get a “system of differential equations”. On its own, this system isn’t super useful. It just tells us that if we know all the specific amounts for a given day, we can predict the amounts on the next day.

However, if we could “solve” this system of equations by turning it into a single formula for how much grass there is over time, one for how many mice there are over time, and one for how many cats there are over time, we can use the initial amounts to reason about what the system will look like at any date in the future, as well as being able to draw graphs of each population to reason about the overall dynamic behavior of the system.

In this case, we may see that each population experiences periodic rises and falls, kind of like a sine wave (but shaped a bit different). For example, when there aren’t many mice, the cats die off and the grass can grow a lot. Fewer cats and more grass are perfect conditions for mice to reproduce quickly, so they will! But eventually, there will be about as many mice as the grass can support, and the cats will start being well-fed, so the cat population will rise. Lots of cats and not much grass is really bad for mice… their numbers will go down now. The other populations (grass and cats) will have similar ups and downs.

There may be an “equilibrium” – an amount of grass, mice, and cats that is somehow “perfect” such that all of the change rates are zero. If we start at the equilibrium levels, we will just stay there forever. But if we start somewhere else, we might see the populations get closer and closer to the equilibrium, or oscillate around it (basically trying to get there but always overshooting it because the other populations aren’t there at the same time), or it may do both (oscillate but get closer to equilibrium over time).

Much of what you learn in differential equations is designed to help you reason about systems like this: dynamic systems where *quantities* and *rates of change* have dependencies which are known, and we want to reason about the long-term behavior of the system.

When you solve a differential equation you essentially figure out what the curve is based on information of what the slope of the curve is.