## Ode Homework Solutions

Faculty: Jerry L. Kazdan Telephone: (215) 898-5109 email: Office Hours: Wed. 10:30-11:30 (and also by appointment) in DRL 4E15 |

TA: Howard Levinson Telephone: (412) 841-8097 email: Office Hours: ??? (and by appointment) in DRL 1N1 |

### Partial Differential Equations, Spring 2015

**Text: **Walter A. Strauss, * Partial Differential Equations: An Introduction*, 2^{nd} Edition, John Wiley (2007), ISBN-13: 9780470054567

As usual, since prices vary considerably, it is wise to search online for less expensive textbook sources.

Note that the first edition had many typos. For a list for both the first and second editions, see the author's web page**Content: ** The heart of this course is to achieve some real understanding of the wave, heat, and Laplace equations. The emphesis will be on mathematical and physical insight and ideas, not complicated formulas.

I taught this in Spring 2011. Although the course will be somewhat different, much of the material will be identical. You might find the homework, exams, and notes from that course useful: Math 425 Spring 2011

Prerequisites & Review Material

Course and Homework Grading

**Some References: **books, articles, web pages**Exams:**

*You may always use one 3"x5" card with*handwritten

*notes on both sides*

**Notes: **

LaTeX: If you will be writing many documents that contain equations, it is wise to learn (and use) LaTeX. It is available on Windows, Macs, and Linux -- and is *free*. See TeX Stuff. For some students, this might be the most useful item you learn in this course.

Some Classical PDEs

Striking a Match: Turbulence

Tacoma Narrows Bridge

Some notes (1965!) from a course like our Math 240 (there are typos.)

ODE's: Generalities on Linear ODE's DeTurck Notes

DeTurck's Math 425 for 2010

DeTurck notes on first order PDE's

Derivation of the heat equation

PDE: Change Variables: print version (display version)

Orthogonal Vectors and Fourier Series**Uniqueness for the initial value problem for the heat equation.** The proof in Petrovsky assumes that the solution u(x,t) is bounded while the proof in John allows for the solution to grow at infinity as long as for any T there is a constant c so that |u(x,t)|< e^{(c|x|2)} for all 0 ≤ t ≤ T. Note that if u(x,t) is allowed to grow too quickly at infinity, there are examples where uniqueness fails.

Standing Waves Music

Gibbs Phenomenon

Lorentz Transformations

Sines and Cosines using ODE

Spherical Harmonics Spherical Harmonics (Wikipedia)

Spherical Harmonics: Strauss

Completeness of Eigenfunctions of the Laplacian. A more "geometric" version of the proof in Strauss, Sec 11.3.

Notes on Convolution

Hear the Shape of a Drum Google search on: Gordon "Shape of a drum"

JLK Australia 2008 Notes (for a slightly more advanced course)

**Homework Assignments:**

- Set 0: Rust Remover (LaTeX source ). Due: Never. This will not be collected.
- Set 1 (LaTeX source ). Due: Thurs., Jan. 22 in class
- Set 2 (LaTeX source ). Due: Thurs., Jan. 29 in class [solutions]
- Set 3 (LaTeX source ). Due: Thurs., Feb. 5 in class [solutions]
- Set 4 (LaTeX source ). Due: Thurs., Feb. 12 in class [solutions]
- Set 5 (LaTeX source ). Due: Thurs., Feb. 19 in class [solutions]
- Set 6 (LaTeX source ). Due: Thurs. Feb 26 in class [solutions]
- Set 7 (LaTeX source ). Due: Thurs., Mar. 5 in class [solutions]
- Set 8 (LaTeX source ). Due: Thurs., Mar. 26 in class [solutions]
- Set 9 (LaTeX source ). Due: Thurs., April 2 in class [solutions]
- Set 10 (LaTeX source ). Due: Thurs., April 9 in class [solutions]
- Set 11 (LaTeX source ). Due: Thurs., April 16 in class [solutions]
- Set 12 (LaTeX source ). Due: Thurs., April 23 in class [solutions]

**Old Exams: ** (you may always use one 3"x5" card with notes on both sides)

Spring 2011 Exam 1, condensed, (solutions).

Spring 2011 Exam 2, condensed (solutions)

## Ordinary Differential Equation MA 3471 D10

Joseph D.Fehribach

Syllabus of Homework Assignments

#### Walter Kelley and Allan Peterson, *The Theory of Differential Equations, Classical and Qualitative*,

#### Pearson Educational, Inc. (Pearson/Prentice Hall), 2004.

1 | 1.1 | pp. 15-19 | Due Friday, 19 March 2010 | #A (below), 1.4, 1.5i,iii, 1.9 | |||

2 | 2.6 | pp. 72-80 | Due Tuesday, 23 March 2010 | #2.3ii, 2.4, 2.5ii, 2.7, 2.18, 2.19ii, 2.22 ^{ } | |||

3 | 2.6 | pp. 72-80 | Due Friday, 26 March 2010 | #2.19i (find e^{At}), 2.24, 2.51i,iii | |||

4 | 2.6 | pp. 72-80 | Due Thursday, 1 April 2010 | #2.54ii, 2.58, 2.59, 2.60 ^{ } | |||

5 | 3.9 | pp. 142-152 | Due Tuesday, 6 April 2010 | #3.1, 3.2iii, 3.5, 3.7i, ii, iii, 3.12 ^{ } | |||

6 | 3.9 | pp. 142-152 | Due Friday, 9 April 2010 | #3.8i, 3.9i, 3.11i, iv, 3.15, 3.20 ^{ } | |||

7 | 3.9 | pp. 142-152 | Due Friday 16 April 2010 | #B (below), 3.37, 3.40, 3.52 ^{ } | |||

8 | 4.4 | pp. 179-184 | Due Tuesday 20 April 2010 | #C (below), 4.2, 4.7, 4.20, 4.28 ^{ } | |||

9 | 8.11 | pp. 386-391 | Due Monday 26 April 2010 | #8.2b,d, 8.4, 8.11, 8.13, 8.21 ^{ } | |||

10 | 8.11 | pp. 386-391 | Due Friday 30 April 2010 | #8.14, 8.15, 8.16, 8.24, 8.25 ^{ } | |||

11 | 5.10 | pp. 264-271 | Due Monday 3 May 2010 | # 5.1ii, iii, 5.15ii, iii, 5.17, 5.18, 5.19, 5.21! ^{ } |

**Color Code Key**

99 Problem whose solution in the back of the text is incomplete or wrong, or whose statement in the text is not precisely correct.

- # A. Please solve and give the maximum interval of existence for
- ( i)
*x*′ = 2*t*/(*t*^{ 2}- 25),*x*(0) = ln(25) - (ii)
*x*′ = tan(*t*),*x*(0) = 0

- ( i)
- # B. Please show that the system (3.46), (3.47) can be transformed to the equivalent polar form we discussed in class:
*r*′ =*r*(1 -*r*^{2}),*θ′*=-1 - # C. Please find the second order solution
*x*for the IVP in Example 4.1, pp. 153-154._{2}

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