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.

Differential Equations

Direction Fields

Definition: Direction Field

A direction field is a graphical representation of all solutions to an ode.

NOTE: Direction fields will be sketched using technology only.

Investigation 05

Use technology to sketch the direction fields for the following odes:

1. ${\displaystyle \frac{dy}{dx}=\frac{y}{x^2+1}}$$\displaystyle \frac{dy}{dx}=\frac{y}{x^2+1}$

2. ${\displaystyle \frac{dy}{dx}=\frac{x}{\sec (x)\sin (y)}}$$\displaystyle \frac{dy}{dx}=\frac{x}{\sec{(x)}\sin{(y)}}$

3. ${\displaystyle (x^2-1)y^{3}dx+x^{2}dy=0}$$\displaystyle (x^2-1)y^3\,dx+x^2\,dy=0$

4. ${\displaystyle x\frac{dy}{dx}=y-3}$$\displaystyle x\frac{dy}{dx}=y-3$

5. ${\displaystyle \frac{dy}{dx}=1+x-y-xy}$$\displaystyle \frac{dy}{dx}=1+x-y-xy$

Use Technology to Sketch Direction Fields.

NOTE: On all Assessments, you may use technology to sketch directions fields without showing any work.

Example 03: Sketch the direction field for: ${\displaystyle \frac{dy}{dx}=\frac{y}{x+1}}$$\displaystyle \frac{dy}{dx}=\frac{y}{x+1}$:

Mathematica:

1
(* Example 04: Sketch direction field for dy/dx=y/(x+1) *)
2
(* The first argument is a grouping of 1 and DE *)
3
(* The last two arguments are the values for the graph to be shown *)
4
StreamPlot[{1, y/(x+1)}, {x, -5, 5}, {y, -5, 5}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator.

3. StreamPlot[ ] has at least three arguments:

   1. The ODE to be sketched, using $\{\text{ and }\}$$\{ \text{ and } \}$ to group

      1. 1; and

      2. the ODE function (NOT the equation)

   2. The interval of $x$$x$-values to be shown in the direction field;

   3. The interval of $y$$y$-values to be shown in the direction field;

4. The dependent variable does NOT use function notation, i.e., $\text{y}$$\text{y}$;

5. For help on using the StreamPlot[ ] function:

   1. In Mathematica, execute the code: $\text{?StreamPlot}$$\text{?StreamPlot}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the StreamPlot[ ] function – you will be able to copy-paste code.

6. Remember, correct Mathematica code will be all black except for variables.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

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

• Section 9.2: # 3, 4, 5, 6, 7 [technology only], 9 [technology only], 13, [technology only].

• Section 9.4: # 3, 7, 9, 11.

• Section 3.8: # 1, 3, 7, 17, 18, 21, 29.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Thursday, 15 October 2020 6:42 EDT Last Modified: Thursday, 14 July 2022 - 10:45 (EDT)