Chapter 9: Introduction to Differential Equations

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Imagine you just jumped out of an airplane, you clear the door, and, well, gravity just grabs you.

Right, it's pulling you down, accelerating you at this constant,

terrifying rate.

Exactly.

But almost immediately, you feel the wind fighting back, the air rushing past you is pushing up.

And the really crucial part is that the faster you fall, the harder that wind pushes back against you.

So how do you predict your exact speed at, say, 12 seconds into the fall?

I mean, you can't really use standard algebra for that.

No, you really can't.

To solve a universe that is, you know, constantly moving and reacting and changing its own rules based on what just happened a millisecond prior.

You need a completely different mathematical language.

It requires a massive conceptual leap.

And if you are the college student listening to this right now, staring down chapter nine of your calculus,

early transcendentals textbook, you are standing on the edge of that exact leap.

Welcome to the deep dive, by the way.

Today we are tackling differential equations.

And it's a big shift, because standard algebra is, it's like taking a photograph.

You isolate a variable, you solve for x, and boom, you find out x equals four.

Just a single static snapshot.

Exactly.

But the material we are diving into today, differential equations, isn't a photograph.

It's a movie.

OK, let's unpack this.

What actually is a differential equation?

If you're seeing this for the very first time, what are you looking at?

Well, essentially, you are looking at an equation made up entirely of rates of change.

It involves an unknown function, which represents some real world system and its derivatives.

And the order of the equation is simply the highest derivative you can spot in the mix.

Right.

And today we are keeping our focus entirely on first order differential equations, which means we are dealing with systems where we only know the first derivative.

The immediate rate of change.

Yes, exactly.

The goal here is a kind of mathematical detective work.

You are given the rules of how something changes over time, and you need to work backward to reconstruct the original function.

But when you integrate something to work backward, there is always a catch.

Any student who has survived Calculus 1 knows about the infamous plus C.

Ah yes, the constant of integration.

Right.

So when you solve a differential equation, that constant means you don't just get one answer, you get a general solution, which is actually this infinite family of parallel curves.

I like to think of it like knowing the general shape of a roller coaster, but not knowing how high off the ground it is.

To pin down the exact roller coaster you are riding, you need an anchor.

Which in mathematics we call an initial condition.

Precisely.

It's a single verified data point.

Like knowing that at time zero, a bacteria population was exactly 500.

So when you pair a differential equation with an initial condition, it creates an initial value problem.

Yes, and you use that specific starting point to lock in the value of C.

That isolates your particular solution, the single exact curve of reality.

So if the whole game is working backward from a rate of change, how do we actually do it?

I mean, let's look at the most intuitive method first.

It's designed for equations where the rate of change is a product of two distinct functions.

One involving only your independent variable, like time,

and one involving only your dependent variable.

Okay, and we call this technique separation of variables, right?

We do.

It relies on a very elegant algebraic manipulation.

You essentially treat your derivative, the UD over, over DX, as a fraction that can just be broken apart.

Wait, you just split the die in DX?

Yep.

The core strategy is to physically move every single term involving a Y to the left side of the equal sign, and every single term involving an X to the right side.

So it's basically like sorting laundry before you run the wash.

That is a perfect way to look at it.

You don't just throw everything in together.

You pull all the colors to the left pile, you toss all the whites to the right pile, and once everything is completely segregated, you can finally process them.

And in this case, processing them just means throwing an integral sign in front of both sides.

Once they're separated, the left side integrates with respect to Y, the right side integrates with respect to X.

And suddenly you have your original function.

That's actually really clean.

It is.

The textbook uses a beautiful real -world example to demonstrate this, actually.

Modeling the thickness of a glacier.

Oh, right.

A glacier is essentially a massively slow -moving river of ice, and its shape is dictated by this massive tug of war.

Exactly.

The immense pressure forces within the ice are fighting against the friction dragging at the rocky base of the mountain.

So when you set up the force balance equation for that glacier,

the rate of change of its thickness, like how steep the ice gets as you walk away from the front edge, is equal to the friction divided by the product of the ice density and gravity.

And here's the kicker.

Friction, density, and gravity are all constants.

So the entire right side of that equation is just one solid, unchanging number.

Which makes the math incredibly easy.

If the rate of change of thickness is just a constant number, separating the variables is trivial.

You just multiply the distance variable over to the right side, integrate both sides, plug in your initial condition.

Which is that the thickness is zero right at the very tip of the glacier.

Right.

And you magically generate an exact parabolic curve, showing the shape of the ice.

What's fascinating here is how this exact same sorting laundry method cracks open exponential growth and decay.

Oh, like a population of bacteria in a petri dish.

Yes.

Consider a system where the rate of change is directly proportional to the amount of material currently present.

If you have 10 bacteria, they reproduce at a certain speed.

But if you have a million bacteria, the sheer number of organisms means the population is just exploding.

The rate of change is tethered directly to the total amount.

Or on the flip side, radioactive decay.

The more uranium you have, the more radiation it emits.

So to solve that, you take the rate equation, divide both sides by the total amount to gather your dependent variables on the left, and multiply your time variable to the right.

When you integrate, something special happens.

The integral of 1 over a variable yields a natural logarithm.

Right.

And to get your final function, you have to use the exponential function, e, to cancel out that logarithm.

And out pops the classic exponential formula.

The amount at any given time equals the starting amount, multiplied by e raised to a constant growth or decay rate.

This is where those infamous concepts of doubling time and half -life come from.

Exactly.

And notice the profound implication of that formula.

The time it takes for a radioactive isotope to decay to half its mass.

Or for a bacteria colony to double in size.

Right.

It completely ignores the initial amount.

It is a fundamental property of the material's specific growth constant.

I love that.

A million cells doubling takes the exact same amount of time as 10 cells doubling.

It's mathematically beautiful.

It is.

But let me push back on this pure exponential model for a second.

Because it assumes the universe is totally limitless.

It assumes the change is always proportional to the total absolute amount.

But real life isn't usually like that.

What if the rate of change is driven not by the total amount, but by the gap between where you are and where you are headed?

That introduces a crucial modification.

We are moving into linear differential equations that include a baseline or a target value.

So mathematically, the rate of change is proportional to the difference between your current state and some fixed constant.

Yes.

And algebraically, if you just treat that entire difference as a single chunk, you can separate the variables exactly like we just did.

So what does this all mean for real life?

How does adding that mathematical target change the physical behavior we observe?

It acts like a magnet, pulling the system toward equilibrium.

The classic example here is Newton's law of cooling.

OK, so if you brew a cup of coffee, its temperature doesn't just plummet infinitely into the negative degrees.

Exactly.

The rate at which it cools is proportional to the difference between the scalding coffee and the ambient temperature of your kitchen.

Ah.

So when the coffee is incredibly hot, the gap between it and the room temperature is massive.

That huge difference drives a very rapid rate of change.

The temperature drops fast.

But as the coffee cools down, that gap shrinks.

And because the gap shrinks, the cooling rate shrinks with it.

The coffee approaches room temperature slower and slower and slower.

It creates a mathematical curve that flattens out, forming a horizontal asymptote right at the room's baseline temperature.

And that is the exact same mechanism governing your skydiver from earlier.

Oh, right.

Gravity applies a constant downward force, but the upward force of air resistance relies entirely on velocity, which perfectly maps to this model.

Yes.

When you first jump, your velocity is zero, so air resistance is zero.

Gravity has total control and you accelerate rapidly.

But as your velocity climbs, the air resistance climbs with it, eating away at your acceleration.

The gap between your current speed and your absolute maximum speed is shrinking.

And eventually, your downward gravitational force and your upward wind resistance balance out completely.

The acceleration hits zero.

And you flatten out at what we call terminal velocity.

Exactly.

This raises an important question, though.

So far, we have been very lucky.

We've looked at equations where we can cleanly separate variables and integrate our way to a perfect, explicit formula.

Right.

But mathematics has a harsh reality.

Many of the differential equations that model complex, real -world systems simply cannot be solved with explicit formulas.

The integrals are impossible.

There is no algebraic trick to save you.

So if we can't solve it mathematically, how do we understand the system?

Do we just give up?

We shift our perspective.

We stop trying to find the formula, and instead, we use the differential equation itself as a literal set of physical instructions.

I see.

Remember, the equation tells us the rate of change the slope at any given set of coordinates.

We could use that to draw a picture of the solution.

This is where slope fields come in.

I find this incredibly cool.

You essentially build a map of the wind.

That's a great way to put it.

You take a piece of graph paper, and for a grid of points all over that paper, you plug the coordinates into your differential equation.

The equation spits out a number, and that number is the slope.

So at every single dot on your grid, you draw a tiny little line segment angled at that slope.

And when you step back and look at the entire grid, you see the flow of the mathematics.

It looks like iron filings tracing a magnetic field or currents in a river.

And even without a formula, you can pick a starting point on that map and simply sketch a line that flows parallel to all those tiny slope segments.

You are tracing the solution curve visually.

And to save college students from getting carpal tunnel drawing a thousand little lines, the text introduces isoclines.

Oh, those are a lifesaver.

You just set your differential equation equal to a specific constant slope, say a slope of two.

Right.

That gives you an algebraic curve where, you know, for a fact, every single slope segment on that curve is angled at exactly two.

It makes building the map way faster.

But visual maps only get us so far, right?

What if an engineer needs actual numerical data for a system they can't solve explicitly?

For that, we rely on numerical approximation, specifically Euler's method.

I always think of Euler's method like trying to navigate through a dense blinding fog using nothing but a compass.

A perfect analogy.

Walk me through how you navigate the fog mathematically.

Well, you start at your initial condition, your known location in the fog.

You check your compass.

In math terms, you plug your current coordinates into the differential equation to find your immediate slope.

Okay.

Now you can't see the true curving path ahead of you, so you just make an assumption.

You assume that your current slope will stay perfectly straight, and you walk forward in a straight line for a predetermined distance.

We call that distance your step size, or h.

And geometrically, you are just using the tangent line to project your position forward.

You take your starting value and you add the change, which is your step size multiplied by your slope.

Right.

So you take your step and you stop.

You are at a new location in the fog.

You check your compass again.

Calculate the brand new slope at this new location, pivot to face that new direction, and walk another straight line of length.

You just chain together dozens of tiny straight line segments to approximate the true bending curve.

It's honestly brilliant.

It is.

But any student using this method must understand the critical tradeoff at its core.

A true solution curve is continuously bending.

Euler's method assumes the path is totally straight for the entire duration of your step size.

So if you choose a large step size, say, walking a mile in the fog before checking your compass,

you will wander completely off the cliff.

Exactly.

If your step size is small, like checking your compass every single inch, your approximated path will hug the true curve incredibly tightly.

You will be highly accurate, but the computational workload skyrockets.

You have to recalculate the slope thousands of times.

It's why numerical methods like this exploded in usefulness once we invented computers to do the tedious stepping for us.

Makes total sense.

So armed with this understanding of how to visualize and estimate behavior, we can revisit a glaring flaw in our earlier discussions.

Let's go back to populations.

OK.

Back to the bacteria petri dish.

Earlier, we praised the exponential growth model for bacteria,

but populations cannot grow exponentially forever.

Exactly.

The universe is not limitless.

An environment has finite food, finite space, finite resources.

Right.

If a population just kept doubling, the mass of bacteria would quickly outweigh the Earth.

We need to install a mathematical braking system.

Which brings us to the logistic differential equation.

It takes the pure exponential growth rate and multiplies it by a regulating factor.

Yes, a fraction that compares the current population to the environment's carrying capacity.

That's the absolute maximum number of individuals that space can sustain without collapsing.

Here's where it gets really interesting.

Let's look at how that braking term physically operates.

The term is 1 minus the ratio of the population to the carrying capacity.

OK, so when a population is tiny, say, 10 rabbits on a massive island that can support 10 ,000,

the ratio of population to capacity is essentially 0.

Which means the braking term is essentially 1 minus 0.

It simplifies to 1.

It has no effect.

The equation behaves virtually identically to pure unrestrained exponential growth.

The rabbits breed like crazy.

But the math is watching.

As that rabbit population skyrockets and gets closer and closer to that 10 ,000 limit,

the ratio of population to capacity approaches 1.

So if you have 9 ,900 rabbits, the ratio is 0 .99.

Your braking term is now 1 minus 0 .99 because a tiny fraction, 0 .01.

And that tiny decimal multiplies against the massive growth rate and utterly crushes it.

The population growth slams into a wall, slowing to an absolute crawl as it nears the ceiling.

The real revelation of the logistic model is understanding its equilibrium solutions.

An equilibrium is a constant state where the rate of change is 0.

The system is basically frozen.

And in this model, there are two.

The first is when the population is 0.

Which makes sense.

0 rabbits means 0 babies.

But that is an unstable equilibrium.

It's fragile.

Yeah, if even one single pregnant rabbit washes ashore on a piece of driftwood, you are no longer at 0.

The population explodes and the system runs away from that 0 baseline.

Exactly.

Now contrast that with the second equilibrium, when the population perfectly equals the carrying capacity.

So if you have exactly 10 ,000 rabbits, the braking term evaluates exactly to 0.

All growth stops.

But this equilibrium is stable.

Yes.

It's self -correcting.

If you accidentally overshoot the capacity and have 11 ,000 rabbits, the math reflects the starvation.

Your population ratio is greater than 1, which makes the braking term negative.

A negative rate of change means the population declines, dying off until it drops exactly back down to the carrying capacity.

The math naturally enforces the limits of reality.

It is a profound piece of modeling.

But we have to face the final boss of this textbook chapter.

Uh oh.

The final boss.

What happens when a system is so complex that the variables are hopelessly tangled?

We are talking about first -order linear equations that completely defy separation.

This is the stuff that gives students nightmares.

They stare at the algebra and no matter how you add, divide or multiply, you absolutely cannot get all the y's on the left and all the x's on the right.

To ground this, let's visualize the classic mixing tank problem.

Imagine an industrial tank holding 600 liters of pure water and a pipe is dumping heavily sugared water into the tank at a rate of 40 liters per minute.

The tank is being stirred vigorously and a drain at the bottom is letting the mixed sugar water out at 20 liters per minute.

So we want a formula for the exact amount of sugar in that tank at any given second.

The basic premise is logical, right?

The rate of change of sugar equals the rate of sugar entering minus the rate of sugar leaving.

The rate entering is easy.

It's a constant concentration multiplied by a constant flow rate.

But the rate leaving is a tangled nightmare.

Because the tank is filling up, 40 liters are coming in but only 20 are leaving.

The total volume of water is swelling by 20 liters every single minute.

Which means the concentration of the sugar water flowing out the bottom is constantly diluting even as the total amount of sugar increases.

Right, so the variable for the amount of sugar, y, is glued to the variable for time, t, inside a shifting fraction.

And if you try to separate those variables algebraically, you will fail.

But the equation fits a very specific structural template.

It's a linear equation.

Okay, so how do we handle it?

For these, mathematicians devised what is arguably the greatest magic trick in calculus,

the integrating factor.

A magic trick that looks terrifying on paper.

The textbook tells you to create a brand new custom function called an integrating factor by taking inner and raising it to the power of an integral of a piece of your equation.

It does look daunting.

Like why on earth would we intentionally complicate a problem by multiplying the entire equation by a custom build monstrosity?

Because we are acting like chemical engineers.

We're dropping a catalyst into the equation to force a reaction.

Wait, what kind of reaction?

Well in Calculus 1 you learn the product rule, how the derivative of two multiplied functions splits into a messy addition of two parts.

Yeah, first times the derivative of the second plus the second times the derivative of the first.

Exactly.

Our tangled mixing tank equation looks suspiciously like the aftermath of a product rule, but it's broken.

It's incomplete.

Oh, I see where this is going.

The integrating factor is the exact missing piece.

When you multiply every single term in your unseparable equation by this custom e to an integral function, the entire left side of your equation suddenly chemically bonds.

It instantly collapses backward into the clean single derivative of a product.

And it folds in on itself.

Precisely.

And once the entire left side of the equation is just one big derivative, the calculus is trivial.

You just lab an integral sign on both sides.

The integral and the derivative on the left annihilate each other, leaving your variables perfectly isolated.

That's wild.

In the mixing tank problem, the terrifying integrating factor simplifies down algebraically to just time plus 30.

Wow.

So you multiply the equation by it, the messy left side collapses into a single neat package, and you just integrate the right side to find your solution.

It bypasses the algebra trap entirely by rewriting the fundamental structure of the problem.

It requires a deep trust in the underlying mechanics of calculus.

But when you execute it, it feels like unlocking a cheat code for reality.

It really is a wild journey to make as a student.

I mean, we started by just trying to sort laundry, physically pushing variables to opposite sides of a room.

Then we looked at how adding a baseline forces systems like cooling coffee or falling skydivers to flatten out against reality.

And we figured out how to survive in the dark without formulas,

using slope fields to map the wind and Euler's method to inch blindly through the fog.

We installed mathematical breaking systems into populations so they wouldn't devour the earth.

And finally, we used integrating factors to force utterly chaotic tangled systems to fold neatly in on themselves.

If we connect this to the bigger picture,

it really forces you to view your environment through a different lens.

How so?

Well, if the universe is ultimately just a massive interconnected collection of shifting rates, fluctuating global temperatures, growing populations, accelerating markets, cooling oceans, are all the seemingly chaotic, unpredictable events in our lives, just complex differential equations, patiently waiting for us to discover the right integrating factor to make sense of them.

Now that is a thought to chew on before your exam.

Thank you so much for exploring this dynamic landscape with us.

On behalf of the Last Minute Lecture team, thank you for listening, catch you next time.