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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 7

Lesson 1: Modeling situations with differential equations

# Writing a differential equation

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.A (LO)
,
FUN‑7.A.1 (EK)
Differential equations describe relationships that involve quantities and their rates of change. See how we write the equation for such a relationship.

## Want to join the conversation?

• What exactly is the k in the equation mean? Sal said it is a proportionality constant, but I couldn't figure out exactly what that meant.
• k in any equation in Mathematics, usually represents a constant. Any constant. Just like the integration constant 'c'!
• How will you know what the relation is without being told like this? Would the relationship have to be experimentally tested and from there create the equation?
• Yes, this information will usually be determined from gathering a few data points to establish a function. In school the relationship is usually given to you at first, but later on you should be able to determine on your own given data points - in this case; speeds for different times.
• At Sal says that option 2 is especially strange. I know it is not the right answer to this question but, is it possible for a function to be inversely proportional to the square of its input. If so, can I get some examples?
• Option 2 describes a function that's inversely proportional to the square of its output, not its input, but yes - both are possible.

For option 2:
S(t) = k / S^2 { which means the same as S(t) = k / [S(t)]^2 }
Multiplying both sides by [S(t)]^2:
[S(t)]^3 = k
S(t) = k^(1/3)
Or, in other words, S(t) is constant for all t.

An example of a function that is inversely proportional to the square of its input would be
S(t) = k / t^2 { which is actually option 3! }
• What does it mean when it says proportional? I don't understand if that means to multiply or divide, it feels incomprehensible.
• When we say x is directly proportional to y, we mean that as x increases, y increases and as x decreases, y decreases. This can be alternatively written as x/y = constant.
Direct proportionality is represented by x ∝ y (or) x = ky where k is a constant. x∝y can be crudely read as x is proportional to y.

Now, when we say x is inversely proportional to y, we mean that as x increases, y decreases and as x decreases, y increases. This can be rewritten as xy = constant.

This is represented by x ∝ 1/y.

NOTE: The above are only 2 examples of proportionality. There are infinite kinds of proportionality. For example, in an adiabatic process (a thermodynamics concept in physics), pressure of an ideal monoatomic gas is inversely proportional to the five-third (5/3) power of volume.
That is, P∝V^(-5/3)
• Should I have known that speed is distance over time? I wouldn't have known that w/o looking it up. I'm pretty far along in my academic career but there's a lot of things I feel like it's assumed that I should have memorized but never did. I remembered there was a relation between speed, distance time but I still had to look it up. It's one thing to know something, it's another to commit things to memory.
• Isn't the derivative of a position function a velocity function?
• How did you come up with inversely proportional being a fraction? I get that if I have something linear like 3x, you make the inverse by changing the numerator and denominator which makes 1/3x.
• Just by definition (see MathWorld): Two quantities y and x are said to be inversely proportional if y is given by a constant multiple of 1/x, i.e. y = c/x for a constant. ...

Weisstein, Eric W. "Inversely Proportional." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverselyProportional.html
(1 vote)
• Why does Sal use k instead of just 1?
(1 vote)
• If two quantities are inversely proportional, then their product is constant.

So, if 𝑦 is inversely proportional to 𝑥, then 𝑥𝑦 = 𝑘
Divide both sides by 𝑥, and we get 𝑦 = 𝑘∕𝑥
• why is 1,2,3 not arbitrary constants ??
(1 vote)
• Because those are specific constants. When we write an expression or equation, we have different 'levels' of variables, where higher levels are evaluated after lower levels.

The lowest level is specific numbers like 1, -3, and π.

Next is arbitrary constants. For example, take a constant function f(x)=c. When we evaluate this function, we understand that the constant c is already a determined number before we being choosing x-values to plug in. If we evaluate it for a certain x, and then want to evaluate it as a different c, we have to first back up, ignore the x-value we chose, and re-select our c. Then we go back and choose our x again.

Next is variables, like the input to f(x)=3x-5, which you're familiar with. And of course, we can insert levels between these or add more on top as a problem calls for it.
(1 vote)
• which are the steps to forming a DE from a equation and what's mean a independent arbitrary constant.
(1 vote)

## Video transcript

- [Instructor] Particle moves along a straight line. Its speed is inversely proportional to the square of the distance, S, it has traveled. Which equation describes this relationship? So I'm not even gonna look at these choices, and I'm just gonna try to parse the sentence up here and see if we can come up with an equation. So they tell us its speed is inversely proportional, to what? To the square of the distance, S, it has traveled. So S is equal to distance. S is equal to distance. And how would we denote speed then, if S is distance? Well speed is the rate of change of distance with respect to time. So our speed would be the rate of distance with respect to time. The rate of change of distance with respect to time. So this is going to be our speed. So now that we got our notation, S is the distance, the derivative of S with respect to time is speed. We can say the speed which is d, capital S, dt, is inversely proportional. So it's inversely proportional, I wrote a proportionality constant, over what? It's inversely proportional to what? To the square of the distance! To the square of the distance it has traveled. So there you go, this is an equation that I think is describing a differential equation, really that's describing what we have up here. Now let's see, let's see what, which of these choices match that. Well actually this one is exactly what we wrote. The speed, the rate of change of distance with respect to time, is inversely proportional to the square of the distance. Now just to make sure we understand these other ones, let's just interpret them. This is saying that the distance, which is a function of time, is inversely proportional to the time squared. That's not what they told us. This is saying that the distance is inversely proportional to the distance squared. That one is especially strange. And this is saying that the distance with respect to time, the change in distance with respect to time, the derivative of the distance with respect to time, dS/dt or the speed, is inversely proportional to time squared. Well that's not what they said, they said it's inversely proportional to the square of the distance it has traveled. So we like that choice.