## Tentative Syllabus  Math 527 -- Fall 2005

All section numbers refer to the course text: Greenberg, Advanced Engineering Mathematics, 2nd edition. The due dates for handing
in assignments will be given in class. THIS SYLLABUS IS SUBJECT TO CHANGE AS THE SEMESTER PROGRESSES; thus you
5.1, 5.2 Course overview; Laplace transforms, introduction. 5.2: 1, 5-8, 10;Solutions
Entrance problems
Entrance problems solutions
5.3, 5.4 Laplace transforms and applications to ode. 5.3: 1, 3b,e,f 10a,c,d;
5.4: 1a,b,d,g,h,j,l,m,q,r,s,v
Hand in 5.4 1(h),(l),(m) Solutions
5.5, 5.6, 4.1, 4.2 Dirac delta functions and Laplace transforms. 5.5: 1a,b,g, 5a,b,d,f
5.6: 1a,c,d,e,i
Hand in: 5.5: 5(f); 5.6: 1(c),(d), (e) Solutions
4.2  Taylor series, radius of convergence. 4.2: 1,3,9
Hand in 1(c), (j); 3(e),(l)
Solutions
4.3  Method of Frobenius. 4.3 1, 2, 3, 6; Hand in 1(b)(n), 2(a), 6(b)
Solutions
4.5, 4.6 Method of Frobenius, continued Fully worked example from lecture 4.3 1, 2, 3, 6; Hand in 1(b)(n), 2(a), 6(b)
4.6 continued, 7.1-2. Bessel and Hankel functions; 4.6: 1-3, 5-7, 12a)-d); Hand in 2, 6(a), 12(b)
Solutions.
7.2--7.3 Phase plane; phase portraits, singular points, stability. 7.2: 1,4,5,10; 7.3: 1. Hand in 7.2: 4(b), 5(d), 10. Solutions.
7.3 Elementary singularities; examples. 7.3: 1, 9, 11. Hand in 9(a), (b): Solutions.
10  7.4 Phase plane applications.
Worked example of phase portrait
analysis of a nonlinear planar system.
7.4: 2(a)-(e), (j)-(n); Hand in 2(c) (e)
Sketch a graph of the trajectories near
each singular point, if you have enough
information to do so. Solutions.
11  7.5 Limit cycles; van der Pol equation; Exam 1 review. 7.5 4. Hand in 4(c). Solutions.
Some review problems
12  October 11! EXAM #1.
13  Handout [pp. 500-519],
Web notes of Prof. Chan
Regular and singular perturbation expansions. Problem in Prof. Chan's notes; Hand this problem in.
Handout: 25.8, 25.9, 25.10
Solutions.
14    Catch up
15; Oct. 20. 9.6--9.10 Introduction to vector spaces; vector spaces of functions;
inner product; orthonormal bases.
9.6: 1, 10-12, 14; 9.9: 11. 12 (b,c). Hand in 9.6: 12, 9.9: 11
Solutions
16; Oct. 25 17.1--17.2 Vector spaces of functions; best approximation. 17.2: 5(a-h), 12(all). Hand in 5 (b),(d), 12(j),(l).
17; Oct. 27. 17.3 Introduction to Fourier series. 17.3: 1, 4(a,c,g,l), 8. Hand in 1, 4(a,c,g,l)
Solutions
18; Nov. 1; 17.4--17.6 Half and quarter range expansions;
Manipulating Fourier series.
17.4 2(a-d); 17.5: 2(a,b), 17.6: 2(a--d)
Hand in 17.4: 2(b,d), 17.5: 2(b), 17.6: 2(a,b,c).
Solutions, 17.4
Solutions, 17.5, 17.6
19; Nov. 3. 17.7, 11.3 Symmetric matrices. Sturm-Liouville theory. 11.3: 1(a,b,e), 15(a,b); 17.7: 1(all), 8, 9(a--c) Hand in 1(c-f), 9(a-c) Solutions, 17.7
20; Nov. 8. 17.8 More Sturm-Liouville theory. 17.8: 2(a--d),4,5; Hand in 2(a,b), 4, 5
Solutions, part I
Solutions, part II
21; Nov. 10. 18.1--18.3 Separation of variables;
application of Sturm-Liouville theory.
18.3: 6(a--h)
22; Nov. 15 Exam II November 15
23; Nov. 17. 18.1--18.3 Separation of variables continued;
Review.
18.3: 4 Solutions, to 18.3, 4,6
24  17.9,17.10, 18.4 Fourier integral and Fourier transform. 17.10: 3, 4, 6(b,f,j,l), 12 Solutions
18.3: 15, 16 (b)(d), 17 (b)
Hand in 18.3: 6 (b); 17.10: 6, 12
25  18.4 Fourier transform method continued. 18.4: 6, 8, 10; 18.3: 19
Hand in 18.4, 10 and 18.3, 19 on Dec. 8
Solutions
26  19.1--19.2 The wave equation.
27  19.1--19.2 The wave equation. 19.2; 5(b),(c), 6, 8
19.4; 2(a), 7
28  Review