306X Homework, Fall 2002

Problems are from Blanchard, Devaney, and Hall, 2nd ed., unless otherwise noted.

[1.1]	 8.	See the discussion above Problem 8!
	 9.
	10. 	c,d only
		In part d, the sample is from the remains of a
		U.S. president. 
		In what year did this person die? Who was it? 
	18.	Explain your answers clearly and concisely in complete
		sentences.

[1.2]	 1.	Hint for (b): see if you can see a constant solution.
	 2.
	 5.	(Solve a separable equation.)
	22.	Also find the solution satisfying y(-2)=3.

[1.3] A qualitative technique: slope fields

8.  First use Maple to make your own slope field for this problem, using
    the same ranges as in the book. Use the Lab 2 assignment to help
    you recall how to use "dfieldplot".
14. Sketch of slope fields from graph of RHS of DE.
15. Matching slope fields with DEs.

[1.4] A numerical technique: Euler's method

1.  Do it by hand.
6.  Use Maple. (You may find this a bit of a challenge until after Lab 3.)
11. Comparison of Euler (on 6) with qualitative analysis.


[1.5] Existence and uniqueness of solutions

2.  Qualitative prediction from limited info.
14. Example of blow-up in finite time.
18. Example of non-uniqueness of solutions.
    Also: (a.ii) Find a constant solution and a solution of the form
    v = A t3, both of which satisfy the IC v(0)=0.

[1.6] Phase line for autonomous DEs
X.   Problem on rhino sparsity.
2.   (Locate equilibria, sketch graph of RHS vs. y, classify the equilibria,
      and sketch the phase line.)
14.  (Sketch several solutions from phase line sketches of Problem 2.)
      Use the following ICs: y(0)=9,7,0,-4 (not the ones in the book).
30,32,34,46. Sketch of phase line from graph of RHS, and vice versa.

[1.7] Bifurcation
3.   But change the - to a +.
     (Location of bifurcation values and sketches of phase lines.)
     Sketch a full bifurcation diagram too.

[1.8] Linear DEs
2.   (General solution.)
14.  (Solution of IVP.)
16.  (General solution with as many integrals done as possible.)
     Note: if you don't have a table of integrals handy, Maple
     might be helpful: syntax for indefinite integral is "int( x^2, x );".


NOTE: This homework involves a significant amount of Maple use. 
Get started right away. Remember that you cannot expect instantaneous 
printouts of your Maple worksheets. 

2.1   2. (Find the equilibria of two 2D systems.)
	 In (ii) use a different DE: divide the whole RHS by 20.
	 Additionally:
	 (a)  Use Maple to make a direction-field picture for each system.
              Choose the scales to include all equilibria.
              


	 (b) Find the equations for the nullclines of each system,
              and sketch them on your direction fields.


[2.2]   8. In (c), use Maple to create the vector field and
           direction field pictures.
	   In (d), hand-sketch solution curves on your direction
           field hard-copy.
	13-16. (Matching phase portraits and time-curves.)
	   



[2.3]  19. (A "triangular" or "partially decoupled" system.) 
         Skip part (d).

[2.4]   2. as follows.
         (i) Do part (a).
         (ii) Use Maple to plot the verified solution for t between -3 and 1.
              Plot x and y together versus t, and y versus x.
	      On the latter plot, put a dot at Y(0).
              Here is how to do plotting in Maple.
              


         (iii) Use Euler's method with a single step of size of 0.5 to
              approximate Y(0.5), for the IVP given. How big are the
              errors in your approximations of x(0.5) and y(0.5)?

[2.5] 2. (Equilibria of the Lorenz system.)

[3.1] 6. (Rewrite system in matrix form.)
      8. (Rewrite system in component form.)
      16(a). (2nd-order equation to first-order system form.)

[3.2]
  2.    (Complete treatment of system with 2 distinct real eigenvalues.)
  12.   (IVP for system with 2 distinct real eigenvalues.)
 
[3.3]
  2.    (Phase portrait for problem 3.2.2.)

[3.4] Complex eigenvalues.
  3.    EXCEPT ... use the matrix -4 6 instead.
                                  -6 2 
        For part (c), do a rough hand-drawn phase portrait.
        Drawing nullclines will help the accuracy of your picture.
        Aslo, hand-sketch the x vs. t graph for one of the solution
        curves in your portrait.
  9.    EXCEPT ... use the matrix -4 6 instead.
                                  -6 2
	Use Maple to draw and superimpose the direction field and
	the solution of the IVP.
        Recall that you can use "display" to combine several plots:
           plot1:=plot(whatever):
           plot2:=plot(whatever):
           with(plots):
           display(plot1,plot2);

  3.    EXCEPT ... use the matrix -2 4 instead.
                                  -2 2

X.  Use Maple's ability to do parametric plots

         plot([x,y,t=whatever..whatever]);

    to make the following picture: a smiley face, with circles for the 
    head and the eyes and a semicircle for the mouth.
    

[3.5]
	2. and 6. (Linear system with repeated eigenvalue.)
	          Skip 2(c). Do 2(d) by hand using eigendirections 
	          and nullclines. Do (e) by hand using portrait from (d).
		  Do 6(c) with Maple.
	18.       (Linear system with a 0 eigenvalue.)


[3.6]
	34. (Damping for "most rapid approach to equilibrium".)
	    Remember that the ultimate approach to equilibrium 
            is governed by the slowest-decaying term in the solution:
	    the term corresponding to the less negative eigenvalue.
[3.7]
	You will have done enough on this in the Lab!

[3.8]	
	18. (Eigenstuff in the Lorenz system.)
	    For (c) (the sketch), I found it convenient to draw my curves
            only in the 3 planes spanned by pairs of eigenvectors.

[4.1] Forced harmonic oscillators

10.     (Solution of an IVP for a nonhomogeneous equation.)
        You may use Maple to help you with flexible guessing.
        You may NOT use Maple's "dsolve" command (except possibly
        to confirm your answer).

38.     (Multi-term RHS.) See discussion in problem 34.


[4.2] Sinusoidal forcing

14.	Use Maple to plot your the solution of the IVP for t from 0 to 20.
	Make both y vs t plot and a phase-plane (y vs. dy/dt) plot.
	Here's how:
	  
          plot(y,t=0..20,color=[red,blue]);
	  yprime:=diff(y,t);
          plot([y,yprime,t=0..20],color=black);

[4.3] Undamped forcing and resonance

12.	Work it by hand.
        Use Maple to plot your the solution of the IVP for t from 0 to 40.
        Make both y vs t plot and a phase-plane (y vs. dy/dt) plot.

16.	(Beats.) Use Maple to make a plot of y vs. t with the cosine coefficient
	=1 and the sine coefficient=2.

[4.4] 	6. (Finding forcing frequency that gives maximum repsonse.)
	   Note: there are 2 cases.
[5.1]
	6. (Linearization of competing species model at the 3 equilibria
            we didn't do in class Mon 10/22.)
            You may, if you wish, use Maple to compute the eigenstuff:
            with(linalg):
            A:=[[1,2],
                [3,4]];
            eigenvals(A);
            eigenvects(A);

[5.2]   (We've done this section earlier.)

[5.3]	X. Show that the function H given in the answer to 
	   Problem 5.3.15 is in fact a conserved quantity
           along the solutions of the (first) system given
           in that problem. 

[5.4]	2. (Liapunov function.)

[6.1]	Laplace transform
	2.  (Laplace transform from the definition.)
	8.  (Inverse Laplace transform.)	
	14. (Harder inversion: pre-manipulation of Y will be helpful.)
	20. (1st-order ODE IVP.)

[6.2]	14.	Hint: recast term with Heaviside function in the form
		ua(t) f(t-a).
		Also: plot your solution with Maple for t in [0,10].

[Davis 11.4]	11.7.8 (Series solution of 2nd order DE at ordinary point.)
		       First 4 non-zero terms.
		11.4.8 (Ditto.) First 4 non-zero terms.

[Davis 11.5]	2. (Locate and classify singular point(s).)
		6. (Find and solve indicial equation.)
		18. Obtain only the first 2 (not 4) non-zero terms
	 	    for each root of the indicial equation.