Differential Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 9.2 – Direction Fields and Euler's Method

• Section 9.3 – Separable Equations

• Section 9.4 – Models for Population Growth

• Section 3.8 – Exponential Growth and Decay

Expected Educational Results

• Objective 15–01: I can identify ordinary differential equations (ODEs).

• Objective 15–02: I can identify the order of ODEs.

• Objective 15–03: I can solve first-order separable ordinary differential equations.

• Objective 15–04: I can solve first-order separable initial value problems.

• Objective 15–05: I can use technology to sketch a direction field for a first-order ordinary differential equation.

• Objective 15–06: I can model population growth and decay with a differential equation.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Population Models

Exponential Growth and Decay

Definition:

The differential form for exponential growth and decay is $\frac{dP}{dt}=kP$$\frac{dP}{dt}=kP$, where $P=P(t)$$P=P(t)$ is the population at time $t$$t$, $k$$k$ is the proportionality constant.

Definition:

The solution form for exponential growth and decay is $P(t)=P_0{e}^{kt}$$P(t)=P_0e^{kt}$, where ${\displaystyle k=\frac{1}{P}\frac{dP}{dt}}$$\displaystyle k=\frac{1}{P}\frac{dP}{dt}$ is the relative growth rate and ${\displaystyle P_0=P(0)}$$\displaystyle P_0=P(0)$.

Predictions Using Population Models

NOTE: When using a population model, you may need to round your answers:

• For units which are continuous aggregates (e.g., mass), then decimal places may be used;

• For units which are non-continuous aggregates (e.g., hundreds [${10}^2$$10^2$], thousands [${10}^3$$10^3$], millions [${10}^6$$10^6$], etc.), then decimal places may be used up to $n$$n$ in ${10}^n$$10^n$;

• For units which refer to individuals (e.g., persons, farmers, wolves, sheep, cells, etc.), then you must round down, a.k.a., truncate, to a whole number.

Investigation 06

In 1960 the total population of Clarkston, Ga was 1524. In 1980 the total population grew to 4539.

1. Find an exponential growth model for the population of Clarkston, Ga.

2. Use the exponential growth model to predict the population of Clarkston, Ga in 2000.

3. Use the exponential growth model to predict the population of Clarkston, Ga in 2010.

4. Use the exponential growth model to predict the population of Clarkston, Ga in 2050. Is this prediction reasonable? Explain.

Investigation 07

In 1960 the total population of Albany, Ga was 55890. In 1980 the total population grew to 74425.

1. Find an exponential growth model for the population of Albany, Ga.

2. Use the exponential growth model to predict the population of Albany, Ga in 2000.

3. Use the exponential growth model to predict the population of Albany, Ga in 2010.

4. Use the exponential growth model to predict the population of Albany, Ga in 2050. Is this prediction reasonable? Explain.

Logistic Model for Population Growth

Definition:

The differential form for the logistic growth model is $\frac{dP}{dt}=kP(1-\frac{P}{M})$$\frac{dP}{dt}=kP\left(1-\frac{P}{M}\right)$, where $P=P(t)$$P=P(t)$ is the population at time $t$$t$, $k$$k$ is the proportionality constant, $M$$M$ is the carrying capacity (i.e., maximum population under the environmental conditions) of the population.

Definition:

The solution form for the logistic growth model is $P(t)={\displaystyle \frac{M}{1+A{e}^{-kt}}},\text{ where }A={\displaystyle \frac{M-P_0}{P_0}}$$P(t)=\dfrac{M}{1+Ae^{-kt}}, \text{ where } A=\dfrac{M-P_0}{P_0}$, where ${\displaystyle P_0=P(0)}$$\displaystyle P_0=P(0)$.

Investigation 08

1. What happens to ${\displaystyle \frac{dP}{dt}}$$\dfrac{dP}{dt}$ when $P$$P$ is small compared to $M$$M$ (i.e., $P<<M$$P<<M$)? Explain. Does this make sense? Explain. NOTE: $<<$$<<$ is the mathematical abbreviation for “much smaller than.”

2. What happens to ${\displaystyle \frac{dP}{dt}}$$\dfrac{dP}{dt}$ when $P$$P$ has a value close to $M$$M$? Explain. Does this make sense? Explain.

Investigation 09

In 1960 the total population of Clarkston, Ga was 1524. In 1970 the total population grew to 3127. In 1980 the total population grew to 4539. Assume M = 20000.

1. Find an logistic growth model for the population of Clarkston, Ga.

2. Use the logistic growth model to predict the population of Clarkston, Ga in 2000.

3. Use the logistic growth model to predict the population of Clarkston, Ga in 2010.

4. Compare your answers to the results of the exponential model.

Investigation 10

In 1960 the total population of Albany, Ga was 55890. In 1970 the total population grew to 72623. In 1980 the total population grew to 74425. Assume M = 800000.

1. Find an exponential growth model for the population of Albany, Ga.

2. Use the exponential growth model to predict the population of Albany, Ga in 2000.

3. Use the exponential growth model to predict the population of Albany, Ga in 2010.

4. Compare your answers to the results of the exponential model.

Newton’s Law of Cooling

Definition:

Let $T$$T$ be the temperature of an object at time $t$$t$. Let $T_s$$T_s$ be the [constant] temperature of the surroundings. Then, the differential form for Newton’s Law of Cooling is

${\displaystyle \frac{dT}{dt}=k\left(T-T_s\right)}$$\displaystyle{\color{red}\frac{dT}{dt}}=k\left({\color{blue}T-T_s}\right)$, $T(0)=T_0$$T(0)=T_0$

Procedure:

Let ${\displaystyle y=T-T_s}$$\displaystyle \color{blue}y=T-T_s$.

Then ${\displaystyle \frac{d}{dt}y=\frac{d}{dt}(T-T_s)\Rightarrow \frac{dy}{dt}=\frac{dT}{dt}}$$\displaystyle \frac{d}{dt}y=\frac{d}{dt}(T-T_s)\Rightarrow \color{red}\frac{dy}{dt}=\frac{dT}{dt}$, and

$T(0)=T_0\Rightarrow y(0)=T(0)-T_s$$T(0)=T_0\Rightarrow y(0)=T(0)-T_s$

Substituting into the differential form of Newton’s Law of Cooling:

${\displaystyle \frac{dy}{dt}=ky}$$\displaystyle{\color{red}\frac{dy}{dt}}=k\color{blue}y$, $y(0)=T(0)-T_s$$y(0)=T(0)-T_s$

Solve for $y(t)$$y(t)$, then convert to $T(t)=y(t)+T_s$$T(t)=y(t)+T_s$.

Compound Interest

Definition:

Let $A_0$$A_0$ be the initial investment; let $r$$r$ be the interest rate (as a decimal); let $n$$n$ be the number of times interest is compounded annually; let $t$$t$ be the number of years of the investment. Then the value of the investment, $A(t)$$A(t)$, is

${\displaystyle A(t)=A_0{(1+\frac{r}{n})}^{nt}}$$\displaystyle A(t)=A_0\left(1+\frac{r}{n}\right)^{nt}$

Compounding Interest Continually

Definition:

Compounding continually implies $n\to \mathrm{\infty }$$n\rightarrow\infty$

So, ${\displaystyle A(t)=\underset{n\to \mathrm{\infty }}{lim}{A}_0{(1+\frac{r}{n})}^{nt}=A_0{e}^{rt}}$$\displaystyle A(t)=\lim_{n\rightarrow\infty}{A_0\left(1+\frac{r}{n}\right)^{nt}}=A_0e^{rt}$.

Homework

At this time, you should be able to complete the following assignments:

• Section 9.4: # 3, 7, 9, 11

• Section 3.8: # 1, 3, 7, 17, 18, 21

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Created: Thursday, 15 October 2020 6:42 EDT Last Modified: Thursday, 14 July 2022 - 16:12 (EDT)