Differential equations, a tourist's guide | DE1

Taking a quote from Stephen Strogatz, since Newton,

mankind has come to realize that the laws of physics are always expressed in the

language of differential equations.

Of course, this language is spoken well beyond the boundaries of physics as well,

and being able to speak it and read it adds a new color to how you view the world around

you.

In the next few videos, I want to give a sort of tour of this topic.

The aim is to give a big picture view of what this piece of math is all about,

while at the same time being happy to dig into the details of specific examples as they

come along.

I'll be assuming you know the basics of calculus,

like what derivatives and integrals are, and in later videos we'll need some basic linear

algebra, but not too much beyond that.

Differential equations arise whenever it's easier

to describe change than absolute amounts.

It's easier to say why population sizes, for example,

grow or shrink than it is to describe why they have the particular values they

do at some point in time.

It may be easier to describe why your love for someone

is changing than why it happens to be where it is now.

In physics, more specifically Newtonian mechanics,

motion is often described in terms of force, and force determines acceleration,

which is a statement about change.

These equations come in two different flavors, ordinary differential equations,

or ODEs, involving functions with a single input, often thought of as time,

and partial differential equations, or PDEs, dealing with functions that have multiple

inputs.

Partial differential equations are something we'll

be looking at more closely in the next video.

You often think of them as involving a whole continuum of values changing with time,

like the temperature at every point of a solid body,

or the velocity of a fluid at every point in space.

Ordinary differential equations, our focus for now,

involve only a finite collection of values changing with time.

And it doesn't have to be time per se, your one independent variable

could be something else, but things changing with time are the

prototypical and most common example of differential equations.

Physics offers a nice playground for us here, with simple examples to start with,

and no shortage of intricacy and nuance as we delve deeper.

As a nice warmup, consider the trajectory of something you throw in the air.

The force of gravity near the surface of Earth causes things

to accelerate downward at 9.8 meters per second per second.

Now unpack what that's really saying.

It means if you look at that object free from other forces,

and record its velocity at every second, these velocity vectors will accrue an

additional small downward component of 9.8 meters per second every second,

we call this constant 9.8 g for gravity.

This is enough to give us an example of a differential equation,

albeit a relatively simple one.

Focus on the y-coordinate as a function of time.

Its derivative gives the vertical component of velocity,

whose derivative in turn gives the vertical component of acceleration.

For compactness, let's write the first derivative

as y-dot and the second derivative as y-double-dot.

Our equation says that y-double-dot is equal to negative g, a simple constant.

This is one we can solve by integrating, which

is essentially working the question backwards.

First, to find velocity, you ask, what function has negative g as a derivative?

Well, it's negative g times t, or more specifically,

negative gt plus the initial velocity.

Notice that there are many functions with this particular derivative,

so you have an extra degree of freedom which is determined by an initial condition.

Now what function has this as a derivative?

It turns out to be negative one-half g times t squared plus that initial velocity

times t, and again we're free to add an additional constant without changing the

derivative, and that constant is determined by whatever the initial position is.

And there you go, we just solved a differential equation,

figuring out what a function is based on information about its rate of change.

Things get more interesting when the forces acting on a body depend on where that body is.

For example, studying the motion of planets, stars,

and moons, gravity can no longer be considered a constant.

Given two bodies, the pole on one of them is in the direction of the other,

with a strength inversely proportional to the square of the distance between them.

As always, the rate of change of position is velocity,

but now the rate of change of velocity, acceleration, is some function of position,

so you have this dance between two mutually interacting variables,

reminiscent of the dance between the two moving bodies which they describe.

This is reflective of the fact that often in differential equations,

the puzzles you face involve finding a function whose derivative and

or higher order derivatives are defined in terms of the function itself.

In physics it's most common to work with second order differential equations,

which means the highest derivative you find in this expression is a second derivative.

Higher order differential equations would be ones involving third derivatives,