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## AP®︎ Calculus BC (2017 edition)

# 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.

## 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.