Differential Equations
# Introduction

A **differential equation** is just an equation involving a function and its derivatives. **Solving** a differential equation means finding the functions that satisfy the equation.

Why are differential equations so important? Because they come up everywhere--in every branch of the sciences. But why do they come up everywhere? That's something to think about while you do this Exploration. One might say the world presents itself in the language of differential equations, like clues to a puzzle, and it's up to us to solve them!

Here is an example of a differential equation:
\[ \frac{dy}{dx} = 6xy.\]
A solution is a function \(y(x)\) whose derivative is \(6xy.\) We'll see in the next chapter how to find such functions. For now, we can *check* whether a function is a solution, by differentiating and seeing whether the equation is satisfied.

Which of the following is a solution to this equation?

In the last example, we only verified a solution; we didn't solve the equation on our own. But you already know how to solve some differential equations, just from having learned calculus! Consider the equation
\[y'' = -y.\]
Solving this equation means finding a function where, when you differentiate it *twice*, you get the negative of what you started with. You actually know several such functions!

Let \(y\) be such a function with \(y(0) = 0\) and \(y\left(\frac{\pi}{2}\right) = 1.\) First, write down a formula for \(y.\) Then, using a calculator, approximately what is \(y(2)?\)

**Hint:** Think about trigonometric functions.

Now that we have a feel for what differential equations look like, let's see how and why they come up in the sciences. Most of this course is focused on solving differential equations, but for the rest of this quiz, we'll just see where they come from.

The most famous differential equation in the world is Newton's second law of motion: \[F = ma\] It says that force equals mass times acceleration. But in what sense is this a differential equation? How is it an equation involving a function and its derivatives?

Let \(y(t)\) give the position of a particle at time \(t.\) Then Newton's second law can be expressed as \(\text{____________}.\)

Of course, the exact differential equation corresponding to \(F = ma\) depends on what the force is.

Suppose an object with mass 1 is falling through the air with a constant force pulling down due to gravity. There's also a force pushing up from air resistance. Suppose we know that the air resistance force is directly proportional to speed. If \(y(t)\) tells us how far the particle has fallen at time \(t,\) which of the following is the differential equation that describes \(y(t)?\) \((\)The \(k\)'s are constants.\()\)

Let's do another example of setting up a differential equation from basic principles. Suppose we're interested in the population of rabbits on an island. Let \(P(t)\) give the rabbit population after \(t\) months of observation. Which of the following is a reasonable description of how \(P\) might grow? (Assume the population has all the food and space it needs and is unaffected by predators.)

I. \(P\) increases at a constant rate.

II. The rate at which \(P\) increases at time \(t\) is proportional to \(P(t).\)

III. \(P\) increases at a rate that's proportional to \(1 - P(t)^2.\)

After the last few questions, we perhaps have a better idea of *why* differential equations come up in nature. It's easier to understand how things change at an instant than to predict where they'll be in an hour.

Basic principles and heuristics tell us about how quantities change instantaneously: how velocity affects acceleration of a falling object, or how the size of an animal population affects its instantaneous growth. These naturally yield statements about rates of change, which can be converted into differential equations.

The goal for the rest of the course will be to develop techniques for solving them.

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