# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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GODE-ENewsletter, http://www.math.hmc.edu/codee, for articles on modeling with ODEs

IDEA (Internet Differential Equations Activities), created by Thomas LoFaro and Kevin Cooper, offers an interactive virtual lab book with models. http://www.sci.wsu.edu/idea

34 Chapter 2

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date_______

Course/Section

Exploration 2.1. ODEs and Their Solutions

1. Where is that constant?

Solution formulas for first-order ODEs often involve an arbitrary constant C, and it can show up in all sorts of strange places in the formulas. Solve each of the following ODEs for y in terms of t and C.

(a) Ó = 1 + sin t (b) Ó = —y/3 (c) y = t/y (d) y = 2t2y/ lny

2. Let’s check out the ODE Architect.

You can see how good the ODE Architect solver is by creating initial value problems for the ODEs of Problem 1 and using the Architect to solve them and graph the solutions. Then compare the solver graphs with those obtained using the solution formula. For example, use ODE Architect to solve and plot the solution of the IVP y = —y/3, y(0) = 1. Then graph the solution y = e—t/3 and compare. To do this, enter the following two equations on the editor screen:

y = —y/3

u = e—t/3

Next enter the initial condition for the ODE, then solve and plot the solution on one of the graphics screens. Use the custom 2D plot tab to overlay the graph of u. Do the graphs match? Repeat with your own initial data for each of the other three ODEs in Problem 1.

36 Exploration 2.1

3. How many1 solutions does this IVP have?

Find formulas for two different solutions for the IVP / = y1/3, y(0) = 0. Which solution does ODE Architect give? Repeat with y = Ó/3, y(0) = 0. [Hint: Is y(t) = 0 for all t a solution?]

4. The effect of a singularity in the differential equation.

The ODE y = y/1 has a singularity at the point (0, 0) because at that point, y/1 = 0/0, which is undefined. Find a formula for all solutions of the ODE. Does the IVP y = y/1, y(0) = 0, have any solutions? Use ODE Architect for y = y/1, y(1) = a, for various positive values of a and then solve backward in time to see what happens as t gets near zero. Explain.

1Usually an IVP has a single solution, but in this Exploration you will see some exceptions. You can find out why by reading about “existence” and “uniqueness” in your text.

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date_______

Course/Section

Exploration 2.2. Slope Fields

1. What happens in the long term?

The following ODEs are given in Screen 2.2 (Experiment 1). Using ODE Architect, describe what the solutions do as t gets very large. Include sketches or printouts of your solution curves and their slope fields.

(a) y = y- 1 (b) / = t/4 (c) / = (y- t)/10

2. More long-term behavior.

Repeat Problem 1 with the following ODEs.

(a) / = ty (b) / = (y2 - 4)/10 (c) / = (y- 3)/5

38 Exploration 2.2

3. Still more long-term behavior.

Using ODE Architect, describe the long-term behavior of the solutions of y = ysin( t + y).

4. Strange solutions.

Make up your own ODEs, especially ones whose solution curves or slope fields form strange patterns. Use ODE Architect to display your results. Describe the long-term behavior of solution curves. Attach printouts of your graphs.

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date_______

Course/Section

Exploration 2.3. The Juggler

Second-order ODEs of the form y" = f(t, y, y) are to be solved in Explorations 2.3 and 2.4. Since ODE Architect only accepts first-order ODEs, we will replace /' = f by an equivalent pair of first-order ODEs. We do this by introducing v = y as another dependent variable:

y = v

v' = f(t, y,v)

1. What goes up must come down.

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