MATH230 Differential Equations Spring 2008 Homework assignment 11 Now due Thursday, 04/15/08

Before you begin, make sure that you have access to Maple, and that you know how to get Maple to draw a phase-plane direction field for a system of two equations, and how to add approximate solutions to it for any chosen initial condition. All this is explained and done in the Maple worksheet posted last Tuesday on our web-site. The worksheet was written using Maple 11. It should work fine in Maple 10 or even 9.5. I have found that some of the options have changed. For example, in my direction fields I like to use rather heavy black arrows instead of the wimpy red "harpoons" Maple chooses. In Maple11, I like the option arrows=smalltwo. But that option does not exist in Maple 10. You may have to play around with this, or simply use what Maple defaults to by not using the options. I do not know how long it will take you, but it is important that you get to know this plot. The applets can be used to draw both (direction field and solutions in phase plane), but you can't do it in one picture, and the pictures are better in Maple because all arrows have the same length...

(1) Last class we thought about a differential equation to model the spring-mass system. We ended up with the following: If x is the displacement of the mass (from equilibrium, x=0) d 2 x k b dx then = - x - will do just that. The positive constants are m (the mass), k (the dt 2 m m dt spring constant), and b (friction parameter). This differential equation can be rewritten as a system of two linear differential equations in he two variables x and v=dx/dt as follows:

dx dx = v = v dt dt , and letting m=1, this changes to the system . dv k b dv = - x - v = - kx - bv dt m m dt

(a) Use Maple to draw several phase-plane-direction-fields for the last system, assuming that the spring constant k=1, for values of b ranging from 0 to 4. You should print the direction fields and label them by their b-values.

(b) Do you see any changes in these direction fields? Describe, first, in terms of the direction fields what these changes are (geometrically), and then what these changes mean in terms of the motion of the spring-mass-system.

(c) Can you recognize a specific feature when b=3, which is not there, or at least harder to recognize in the other direction fields? Which feature? What does this feature mean in terms of the spring-mass-system? (2) In this exercise we will explore the system that was mentioned in class as describing the motion of a swaying skyscraper (take a look at the handout(s) for pictures):

dx = v dt dv = - x - v + x 3 dt

The variable x is the displacement of the top of the building (sideways from equilibrium), and v is the speed of the movement. I may have more to say about WHY this system is a "reasonable over-simplification" to model this - for now just assume that it is. (The coefficient -1 of x is something like a spring constant, and the term "-v" is a friction term (assumed to depend on the velocity). I am still working on explaining the x3-term. (This is the extra credit opportunity I mentioned in class.)

(a) Draw a coordinate system for a phase plane. Label the axes. Find all equilibrium solutions of this system. Mark them in the phase plane.

(b) Next, we want to characterize the equilibrium points using one of the following: center, spiral sink, spiral source, sink (no spiraling), source (no spiraling), saddle (which means "none of the above" or a "mix of the above"). I may be forgetting some... but these should be enough for now. To characterize the equilibria, you need to examine what happens to solutions that start near the equilibria. Use Maple to study this. You may restrict your phase-plane- direction-field to a region around each equilibrium, and include a whole bunch of initial conditions to have Maple draw the approximate solutions for these initial conditions. Once you feel you have drawn enough solutions to make a decision about how to call the equilibrium, then show the picture, choose the name, and explain in a few words what it was that helped you decide.

(c) Finally, through further investigation, try to figure out for which initial conditions the skyscraper will "recover", and for which initial conditions it will tumble down (not recover.) Explain in suitable pictures and words how you came to your conclusion. (One way to summarize these results is to draw a phase plane, and divide it into regions of "recovers" and "fails to recover". Any further commenting will be valued.