Differential Equations - Word Problem

• Quincy
In summary, the conversation discusses two problems related to the theory of learning and memorization. The first problem involves finding the differential equation for the amount of material memorized over time, assuming that the rate of memorization is proportional to the amount left to be memorized. The second problem takes forgetfulness into account, and the resulting differential equation includes a negative value to account for the decrease in the rate of memorization due to forgetting.
Quincy

Homework Statement

A pond initially contains 1,000,000 gal of water and an unknown amount
of an undesirable chemical. Water containing 0.01 gram of this chemical per
gallon flows into the pond at a rate of 300 gal/hour. The mixture flows out at
the same rate, so the amount of water in the pond remains constant. Assume
that the chemical is uniformly distributed throughout the pond.
a) Write a di erential equation for the amount of the chemical at any time.
b) How much of the chemical will be in the pond after a very long time?
Does this limiting amount depend on the amount that was present initially?

The Attempt at a Solution

t (hours)
y (grams)

y(0) = y0

dy/dt = 3g/hr - (??) -- How do I determine the rate at which it's flowing out if I don't know the initial value?

It says the mixture flows out at the same rate as it flows in such that the volume remains constant (1,000,000 gal). Now just remember that (like you did with the flow in) the flow out is going to be equal to the rate (300gal/hour) multiplied by the concentration of the unknown chemical. Since the volume is constant and the mixture is perfectly mixed you can conclude that the concentration flowing out would be equal to y/V.

Homework Statement

1. In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. If M denotes the total amount that is to be memorized and A(t) the amount memorized in time t, a nd the di fferential equation for A.

2. In Problem 1, assume that the amount of material forgotten
is proportional to the amount memorized at time t. What is the differential
equation for A when forgetfulness is taken into account?

The Attempt at a Solution

1. dA/dt = 1/M -- should I include an arbitrary rate constant?

2. dA/dt = 1/M - A

#1:
It says that the rate at which the material is memorized is proportional to the amount that is left to be memorized. So this could be restated as

(Rate of Memorizing Material) $$\propto$$ (Total Amount to Be Memorized) - (Amount Memorized)

So, since A(t) is the amount memorized at time t, and M is the total amount to be memorized, you could say that

$$\frac{dA(t)}{dt}$$ $$\propto$$ M - A(t)

or like you mentioned we could use a proportionality constant, let's say "c", so

$$\frac{dA(t)}{dt}$$ = c(M - A(t))

That is my guess for an answer for this question. I solved the differential equation and got a solution that made sense, you could do the same to see what I'm talking about.

#2:

For this one we take forgetfulness into account. This means that to our previous differential equation,
$$\frac{dA(t)}{dt}$$ = c(M - A(t))

,we must introduce a negative value on the right hand side. That is because forgetting decreases the rate of memorization. If we denote the forgetfulness as F, then using what they gave us we know that F $$\propto$$ A ,or, if we again introduce a constant of proportionality, F = kA

So the new differential equation we would get is
$$\frac{dA(t)}{dt}$$ = c(M - A(t)) - kA

I also solved this one and it made sense. If you want to try solving these go for it (it's good practice and a great way to check your answer).

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It represents a relationship between a quantity and its rate of change.

2. How are differential equations used in real-life applications?

Differential equations are used to model and understand various physical phenomena, such as motion, heat transfer, population growth, and electrical circuits. They are also used in various fields like engineering, physics, economics, and biology.

3. What are some common methods for solving differential equations?

The most commonly used methods for solving differential equations include separation of variables, substitution, integrating factors, and numerical methods such as Euler's method and Runge-Kutta methods.

4. How can I tell if a word problem involves a differential equation?

If a word problem involves describing a relationship between a quantity and its rate of change, then it likely involves a differential equation. Look for keywords such as "rate," "change," "derivative," or "differential" in the problem.

5. Are there any real-life examples of differential equation word problems?

Yes, there are many real-life examples of differential equation word problems. Some examples include predicting the spread of an infectious disease, modeling drug concentration in the body, and determining the motion of a falling object subject to air resistance.

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