Math 424 Sp07

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Math 424: Ordinary Differential Equations

Instructor: Steve Rosencrans
Office: Gibson Hall 427
Office Hours: MWF at 2PM or by appointment (see me after class or phone or email for appointment)
Phones: 862-3447 (office) or 865-5727 (math office) or 782-6912 (cell)
Email: srosenc@tulane.edu

Differential Equations, Brannan and Boyce.

Software: The Student Edition of Matlab, version 7 or earlier.

Grade	Weight
Final Exam	45%
Tests (2)	35%
Projects (2)	20%

Projects: There will be 2 group projects. Groups of 3 will be created.

MATLAB: This course makes use of MATLAB, a program for mathematical computation, analysis, visualization, and algorithm development. You should have your own copy. The Student Version (running under Windows or Linux or Mac OS) is reasonably priced (about $110 online at http://www.mathworks.com/products/education/student_version/sc/index.shtml.
You will need three auxiliary files, which you obtain at http://math.rice.edu/~dfield ,

· Dfield7.m

· Pplane7.m

· Odesolve.m

(Be sure that the versions of the pplane and dfield files match your version of MATLAB. If you are not buying a new copy, the files might be dfield6.m etc.) The three files go in the directory toolbox\local of the MATLAB directory on your computer.

Homework: Homework is assigned, and will be discussed at the beginning of each class. It will not be collected.

Homework

1.1	21, 22
1.2	11, 12

1.4	6
2.1	6,7,9,11,20,21,32,33
	Solve y’+y=exp(-t^2), y(0)=1
	Use Euler’s method to estimate y at .01, .02, … , .05
2.2	1,2,22,24,27 use dfield as an aid

29 JAN	1. Find out how to make a function m-file in Matlab. Read about ODE45. Run ODE45 on dx/dt=4x-6sin(t) , x(0)=1, for 0<t<3. Plot the solution. (Don’t worry if you have trouble here, we’ll go over it in class.) 2. Suppose dx/dt < sin(t)x, x(0)=1. What can you say about x(t) for t>0? 3. (Under suitable assumptions about f – say what you’re assuming) show that the solution to dx/dt=f(t,x) depends ‘continuously’ on the initial value, i.e., if x(0) changes by a small amount delta, then the two solutions remain close for a time.
3.1	13,18,26
3.2	1-8,9-14 use pplane
3.3	1-3,12,17,18
6 Feb	In all the following sketch phase portrait by finding formulas for the phase curves (find parametric formulas and eliminate t, or eliminate t at the outset by chain rule dx2/dx1=f2/f1). Confirm by pplane. 1. x’=x y’=y 2. x’= - x y’ = - y 3. x’=2x y’=y 4. x’= -x y’=y 5. x’=y y’=x 6. x’= - 2 x y’=y
12 Feb	3.4: 1, 2, 8, 9, 13 Show that if A has complex conjugate eigenvalues, the solutions of x’=Ax are of the form f1(t)*f2(t) where f1 is a scalar exponential function of t (possible a constant) and f2 is a vector valued function of t which traces out an ellipse.
14 Feb	Prove det(exp(A))=exp(trace(A)) . (Assume A is similar to a diagonal matrix, i.e., there exists an invertible P such that P^{-1}AP is diagonal.) Learn how to use ode45 for 2x2 systems. Apply to x’=y, y’=-x. Verify that the output is periodic. (Why should it be?)

17 Feb	Do the problem assigned in class (verifying formula for solution of x’=Ax in case A has only one eigenvalue, and the corresp eigenspace is 1-D). Prove char’c polynomial of A is \lambda^2 - \lambda tr(A) + det (A). (I left out this step in class when I was proving det(exp(A))=exp(tr(A)).) Solve x’=Ax where A=[1,1;0,1] and x(0)=[1;2]
28 Feb	7.2 1-5
5 March	1-3, 5-9 in notes on generalized functions
14 March	p.221 8 p.242 1-5,9,11 [only (a) and (c) for these], 34 p.250 4,5,11,12,17,30 p.271 1,2,8
20 March	Forced linear oscillator. Assume (non-dimensionalized) as in class x’’+\alpha x’ + x = A cos(\omega t) where \alpha is small and positive. Ignoring transients (terms exponentially small in time), show that x(t) is B * cos(\omega t – psi) where B>0 (depending upon \omega) is the amplitude of the forced motion. Show that as the forcing frequency \omega varies over all real numbers, B(\omega) has a single maximum. What is that maximum and at what critical value of \omega does it occur? (Restore dimensions in the final result.)
3/31	1. If A is a skew-symmetric matrix, A*=-A then the flow of x’=Ax is confined to a sphere about the origin. 2. In class we worked out the explicit formula for solution of initial-value problem for x’=Ax where A is a symmetric matrix. Supply details for special case A = [-2,1;1,-2]. 3. Let D = d/dx. Show that the operator x D^2 + D , on interval 1<x<2 with Dirichlet boundary conditions, is symmetric. 4. (Continued) Show that D itself is skew-symmetric. 5. Show that the operator d^2/dx^2+d^2/dy^2 with Dirichlet boundary conditions on a nice 2D domain is symmetric. (First define the appropriate inner product.)
4/6	1. How would you solve heat equation on a rod 0<x<1 with nonzero (and constant in time) temperature assigned at the boundaries? 2. Show that for t>0, exp(-x^2/(4t))/sqrt(4 \pi t) is a solution to the heat equation on the entire real line, and has initial value delta(x). Interpret probabilistically. 3. Find Neumann eigenvalues and efns for d^2/dx^2 (i.e., boundary conditions are now u_x=0 at the boundaries).
4/8	4. What conditions on a,b,c ensure that ax^2+bxy+cy^2 is positive definite? 5. Show that ax^3+bx^2y+cxy^2+dy^3 is neither pos def nor neg def. 6. Use Lyapunov fns to show stability of the origin for x’=-3x^3-y, y’=x^5-2y^3 7. Use Lyapunov fns to show stability of the origin for x’=-2x+xy^3, y’=x^2y^2-y^3 8. Prove that the origin is unstable for x’=F(x) if there exists function E such that E=0 at origin, every disk about the origin contains at least one point where E>0, and <grad E, F> is pos def. 9. Prove origin unstable for x’=2xy+x^3, y’=-x^2+y^5.
7.5(Poincare)	8, 9, 10, 11, 15, 16cd
Series I	If u satisfies u’’-u=1 find the powers series expansion of an arbitrary solution about x=0. (‘=d/dx)

Math 424: Ordinary Differential Equations

Homework

1.1

21, 22

1.2

11, 12

1.4

6

2.1

6,7,9,11,20,21,32,33

Solve y’+y=exp(-t^2), y(0)=1

Use Euler’s method to estimate y at .01, .02, … , .05

2.2

1,2,22,24,27 use dfield as an aid

29 JAN

1. Find out how to make a function m-file in Matlab. Read about ODE45. Run ODE45 on dx/dt=4x-6*sin(t) , x(0)=1, for 0<t<3. Plot the solution. (Don’t worry if you have trouble here, we’ll go over it in class.)

2. Suppose dx/dt < sin(t)*x, x(0)=1. What can you say about x(t) for t>0?

3. (Under suitable assumptions about f – say what you’re assuming) show that the solution to dx/dt=f(t,x) depends ‘continuously’ on the initial value, i.e., if x(0) changes by a small amount delta, then the two solutions remain close for a time.

3.1

13,18,26

3.2

1-8,9-14 use pplane

3.3

1-3,12,17,18

6 Feb

In all the following sketch phase portrait by finding formulas for the phase curves (find parametric formulas and eliminate t, or eliminate t at the outset by chain rule dx2/dx1=f2/f1). Confirm by pplane.

1. x’=x y’=y

2. x’= - x y’ = - y

3. x’=2x y’=y

4. x’= -x y’=y

5. x’=y y’=x

6. x’= - 2 x y’=y

12 Feb

3.4: 1, 2, 8, 9, 13

Show that if A has complex conjugate eigenvalues, the solutions of x’=Ax are of the form f1(t)*f2(t) where f1 is a scalar exponential function of t (possible a constant) and f2 is a vector valued function of t which traces out an ellipse.

14 Feb

Prove det(exp(A))=exp(trace(A)) . (Assume A is similar to a diagonal matrix, i.e., there exists an invertible P such that P^{-1}AP is diagonal.)

Learn how to use ode45 for 2x2 systems. Apply to x’=y, y’=-x. Verify that the output is periodic. (Why should it be?)

17 Feb

Do the problem assigned in class (verifying formula for solution of x’=Ax in case A has only one eigenvalue, and the corresp eigenspace is 1-D).

Prove char’c polynomial of A is \lambda^2 - \lambda tr(A) + det (A). (I left out this step in class when I was proving det(exp(A))=exp(tr(A)).)

Solve x’=Ax where A=[1,1;0,1] and x(0)=[1;2]

28 Feb

7.2 1-5

5 March

1-3, 5-9 in notes on generalized functions

14 March

p.221 8

p.242 1-5,9,11 [only (a) and (c) for these], 34

p.250 4,5,11,12,17,30

p.271 1,2,8

20 March

Forced linear oscillator. Assume (non-dimensionalized) as in class

x’’+\alpha x’ + x = A cos(\omega t) where \alpha is small and positive.

3/31

1. If A is a skew-symmetric matrix, A*=-A then the flow of x’=Ax is confined to a sphere about the origin.

2. In class we worked out the explicit formula for solution of initial-value problem for x’=Ax where A is a symmetric matrix. Supply details for special case A = [-2,1;1,-2].

3. Let D = d/dx. Show that the operator x D^2 + D , on interval 1<x<2 with Dirichlet boundary conditions, is symmetric.

4. (Continued) Show that D itself is skew-symmetric.

5. Show that the operator d^2/dx^2+d^2/dy^2 with Dirichlet boundary conditions on a nice 2D domain is symmetric. (First define the appropriate inner product.)

4/6

1. How would you solve heat equation on a rod 0<x<1 with nonzero (and constant in time) temperature assigned at the boundaries?

2. Show that for t>0, exp(-x^2/(4t))/sqrt(4 \pi t) is a solution to the heat equation on the entire real line, and has initial value delta(x). Interpret probabilistically.

3. Find Neumann eigenvalues and efns for d^2/dx^2 (i.e., boundary conditions are now u_x=0 at the boundaries).

4/8

4. What conditions on a,b,c ensure that ax^2+bxy+cy^2 is positive definite?

5. Show that ax^3+bx^2y+cxy^2+dy^3 is neither pos def nor neg def.

6. Use Lyapunov fns to show stability of the origin for x’=-3x^3-y, y’=x^5-2y^3

7. Use Lyapunov fns to show stability of the origin for x’=-2x+xy^3, y’=x^2y^2-y^3

8. Prove that the origin is unstable for x’=F(x) if there exists function E such that E=0 at origin, every disk about the origin contains at least one point where E>0, and <grad E, F> is pos def.

9. Prove origin unstable for x’=2xy+x^3, y’=-x^2+y^5.

7.5(Poincare)

8, 9, 10, 11, 15, 16cd

Series I

If u satisfies u’’-u=1 find the powers series expansion of an arbitrary solution about x=0. (‘=d/dx)