# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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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 y = y1/3, y(0) = 0. Which solution does ODE Architect give? Repeat with y = y2/3, 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.

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

2. More long-term behavior.

Repeat Problem 1 with the following ODEs.

(a) / = ty (b) y = (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.

Use ODE Architect to find the position of the ball at several different times t for several different initial velocities. Assume no air resistance and that the ball moves in a vertical line. What is the name for the shapes of the solution curves in the ty-plane? Does it take longer for the ball to rise or to fall? Show and explain the difference (if there is one!).

2. Hand-to-hand motion of the ball.

For a given initial speed vo, find the range of values of the angle do so that the ball goes from one hand to the other. Now increase the initial speed. What happens to the range of successful values of do? Explain. [Suggestion: First take a look at Screen 3.5 (Experiment 2 in Module 2); then enter the equations in ODE Architect and vary do with fixed vo to find the ranges. You may also want to take a look at Screens 1.2 and 1.3 in Module 5.]

40 Exploration 2.3

3. Raise your hand!

Suppose the juggler raises his catching hand one foot higher. Repeat Problem 2 in this setting.

4. Juggling two balls.

Construct model ODEs for tossing two balls, one after the other, from one hand to the other. Use ODE Architect to find the positions of both balls at time t.

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 2.4. The Sky Diver

Second-order ODEs of the form y" = f(t, y, ?) are to be solved in Explorations 2.3 and 2.4. Since ODE Architect only accepts first-order ODEs, we will replace y" = f by an equivalent pair of first-order ODEs. We do this by introducing v = / as another dependent variable:

y = v

v' = f(t, y,v)

1. Terminal speed of a falling body.

Use ODE Architect and determine the sky diver s terminal speeds for several different values of the viscous damping coefficient (use m = 5 slugs and g = 32 ft/sec2). Is there any difference if the sky diver jumps at 25,000 feet instead of 13,500 feet? [Suggestion: After entering the ODE and solving, click on a Data tab in either of the two graphics windows and use approximate data you find there.]

2. Slow down!

If a sky diver can survive a free-fall jump only if she hits the ground at no more than 30 ft/sec, what values of the viscous drag coefficient k make this possible? Are these k-values realistic? (Use m = 5 slugs and g = 32 ft/sec2.)

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