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### Course: Integral Calculus>Unit 2

Lesson 1: Differential equations introduction

# Writing a differential equation

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)
• Why does Sal use k instead of just 1?
• 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
• for the speed, can we represent it as S/t here? if no, why not
(1 vote)
• you can though it is done more in non-rigorous setting.

You can think of ds/dt as fraction where both ds, dt are infinitesimally small numbers. Technically speaking, there some issues with having infinitesimally numbers but you can ignore this.

The key point here is we taking the "difference in the distances" and dividing that by the "difference of time".

This is not convey in S/t. So give example let a 100m rate in 20 seconds.

Than the average speed is (100-0)/(20-0). We take difference across two points (0, 0) and (20, 100)
• Isn't the derivative of a position function a velocity function?
• Yes but I assume they don't want to add too much confusion of the wording.
• How did he get k?