Differential Equations and Linear Algebra (427J)

Course information:

Unique: 54005

Lecture: 11:00-12:30pm, T Th, CPE 2.214

Textbook: Martin Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics, 4th edition ; ISBN-13: 978-0387978949

Instructor: Arie Israel (arie AT math DOT utexas DOT edu)

Office hours: 9:00am-12:00pm, W, RLM 12.126

Discussion session: 8:00-9:00am, M W, RLP 0.126

TA: Yan Zhou (yzhou AT math DOT utexas DOT edu)

TA office hours: 10:00-12:00pm, M, RLM 13.152

Linear Algebra:

Supplementary Notes

Study Tips

First day handout:

Schedule:

• Lecture 1 (08/30): Braun 1.1, 1.2
• Lecture 2 (09/04): Braun 1.4
• Lecture 3 (09/06): Braun 1.9
• Lecture 4 (09/11): Braun 1.10, Braun 2.1
• Lecture 5 (09/13): Braun 2.2
• Lecture 6 (09/18): Braun 2.3, 2.4
• Lecture 7 (09/20): Braun 2.5
• Midterm Exam 1 (09/24): Braun chapter 1 + sections 2.1-2.5
• Lecture 8 (09/25): Braun 2.6
• Lecture 9 (09/27): Braun 2.6
• Lecture 10 (10/02): Braun 2.8
• Lecture 11 (10/04): Braun 2.8
• Lecture 12 (10/09): 3.1, Linear Algebra 1
• Lecture 13 (10/11): Linear Algebra 2
• Lecture 14 (10/16): Linear Algebra 3
• Lecture 15 (10/18): Linear Algebra 4
• Lecture 16 (10/23): Linear Algebra 5
• Lecture 17 (10/25): Linear Algebra 6
• Lecture 18 (10/30): Linear Algebra 7
• Midterm Exam 2 (10/31): Braun sections 2.6, 2.8, 3.1, 3.2, 3.3, 3.7; supplementary linear algebra notes.
• Lecture 19 (11/01): Linear Algebra 8
• Lecture 20 (11/06): Linear Algebra 9, Braun 3.8
• Lecture 21 (11/08): Braun 3.8, 3.9
• Lecture 22 (11/13): Braun 3.9
• Lecture 23 (11/15): Braun 3.10, 3.11
• Midterm Exam 3 (11/19): Braun sections 3.4, 3.5, 3.6, 3.8, 3.9, 3.10, 3.11; supplementary linear algebra notes.
• Lecture 24 (11/20): Braun 5.1, 5.2
• Lecture 25 (11/27): Braun 5.3, 5.4
• Lecture 26 (11/29): Braun 5.4, 5.5
• Lecture 27 (12/04): Braun 5.6
• Final Review (12/06)
• Final Exam (12/15): 7-10 PM; Location: CPE 2.214

Quiz & Exam Solutions:

• Quiz 1
• Quiz 2
• Quiz 3
• Exam 1 (v1); Exam 1 (v2)
• Exam 2 (v1)
• Exam 3 (v1)
• Quiz 4 (v1); Quiz 4 (v2)
• Quiz 5 (v1); Quiz 5 (v2)
• Quiz 6 (v1); Quiz 6 (v2)
• Quiz 7 (v1); Quiz 7 (v2)
• Quiz 8 (v1); Quiz 8 (v2)
• Quiz 9 (v1); Quiz 9 (v2)
• Quiz 10 (v1); Quiz 10 (v2)

Lecture 1: First Order Linear (section 1.2)

Homework:

• Braun, Section 1.2: (page 9) 1, 3, 4, 6, 11, 13, 15, 16 (solutions)
• faculty.sfasu.edu/judsontw/ode/html/firstlook05.html

Exercises 1-20

Challenge:

• Braun, Section 1.2: 17, 18, 19

Notes:

• .txt
• handwritten

Mathlets:

• Isoclines
• Euler's method

Lecture 2: Separable Equations (section 1.4), Exact Equations (section 1.9)

Homework:

• Braun, Section 1.4: (page 24) 1, 2, 3, 4, 6, 8, 9 (solutions)
• faculty.sfasu.edu/judsontw/ode/html/firstlook02.html

Exercises 2-9, 14-20

Challenge:

• Braun, Section 1.4: 11, 13-18

Notes:

• .txt
• notes

Lecture 3: Exact Equations cont'd (section 1.9)

Homework:

• Braun, Section 1.9: (page 66) 2, 4, 5, 6, 7, 8, 9, 12, 13 (solutions)

Challenge:

• Braun, Section 1.9: 15, 16, 17

Notes:

• .txt
• notes

Lecture 4: Existence & Uniqueness Theorem (section 1.10), Second Order Linear Equations (section 2.1)

Homework:

• Braun, Section 2.1: (page 136) 1 - 7 (solutions)

Challenge:

• Braun, Section 2.1: 12, 15

Notes:

• .txt
• notes

Just for fun:

• Blue-Eyed Islanders Puzzle

Lecture 5: Second Order Constant Coefficient Linear Equations (section 2.2)

Homework:

• Braun, Section 2.2: (page 140) 1, 2, 4, 5, 6, 8, 9, 10, 12; (page 149) 1, 4, 5, 6 (solutions)

Challenge:

• Braun, Section 2.2: (page 149) 11, 12, 13, 17, 18

Modeling:

• Harmonic Oscillators
• Cranes and Duffing's Equations
• RLC Circuits

Notes:

• .txt
• notes

Lecture 6: Second Order Constant Coefficient Linear Equations cont'd (section 2.2), Inhomogeneous Equations (section 2.3), Variation of Parameters (section 2.4)

Homework:

• Braun, Section 2.2: (page 144) 1, 2, 5, 6, 8, 9, 13 (solutions)
• http://faculty.sfasu.edu/judsontw/ode/html/secondorder01.html#exercises-secondorder01

Exercises 1-20

Challenge:

• Braun, Section 2.2: (page 140) 9; (page 144) 14, 15, 17, 18

Notes:

• .txt
• notes

Lecture 7: Variation of Parameters (section 2.4), Judicious Guessing (section 2.5)

Homework:

• Braun, Section 2.3: (page 152) 1, 2, 3
• Braun, Section 2.4: (page 156) 1, 2, 3, 5, 6, 9
• Braun, Section 2.5: (page 164) 1, 3, 4, 5, 6, 14 (solutions)

Challenge Problems:

• Braun, Section 2.5: 7, 9, 11

Notes:

• notes

Lecture 8: Judicious Guessing (section 2.5), Mechanical Vibrations (section 2.6)

Homework:

• Braun, Section 2.5: (page 164) 11
• Braun, Section 2.6: (page 172) 1, 2, 3, 4, 5

Challenge Problems:

• Braun, Section 2.6: 7, 8

Notes:

• notes

Lecture 9: Mechanical Vibrations (section 2.6), Power Series (section 2.8)

Notes:

• notes

Lecture 10: Power Series I (section 2.8)

Homework:

• Braun, Section 2.8: (page 197) 1-11, 12 (a) (solutions)

Notes:

• notes

Lecture 11: Power Series II (section 2.8)

Worked Examples

Notes:

• notes

Lecture 12: Systems of ODEs (section 3.1)

Homework:

• Braun, Section 3.1: (page 271) 1,2,5,6,9,10,11,13,14,15 (solutions)

Notes:

• notes

Lecture 13: Gaussian Elimination

Homework:

• pdf

Lecture 14: Reduced Row Echelon Form

Homework:

• pdf (solutions)

Notes:

• notes

Lecture 15: Vector Spaces, Subspaces

Homework:

• Braun, Section 3.2: (page 278) 1-12 (solutions)

Notes:

• notes

Lecture 16: Linear Independence, Span, and Basis

Homework:

• Braun, Section 3.3: (page 288) 1-11, 14 (solutions)

Notes:

• notes

Lecture 17: Linear Transformations and Null Space

Note: This is the last lecture which will be covered on Midterm Exam #2 (October 31).

Homework:

• Braun, Section 3.7: (page 332) 4-9, 13, 17, 19, 20, 21
• Hint: For problem 13: Apply the usual Gaussian Elimination Procedure to the augmented matrix of the system. At the end of the procedure, the final column may contain the variable lambda in several positions. Then it is straightforward to determine the equation lambda must satisfy in order that the system is consistent.

Notes:

• notes

Midterm Exam #2 Review Sheet:

Review Sheet

Review Sheet Solutions

Lecture 19: Determinants and Invertibility

Homework:

• Braun, Section 3.5: (page 308) 5, 6, 8 (solutions)
• additional problems

Notes:

• lecture 19

Lecture 20: Invertibility and Matrix Algebra

Homework:

• Braun, Section 3.6: (page 320) 9, 10, 11, 12 (solutions)

Notes:

• lecture 20

Lecture 21: Systems of ODEs I: Real Eigenvalues

Homework:

• Braun, Section 3.8: (page 340) 1, 2, 3, 4, 6, 7, 8, 10, 11 (solutions)

Notes:

• lecture 21

Lecture 22: Systems of ODEs II: Real Eigenvalues cont'd, Complex Eigenvalues

Homework:

• Braun, Section 3.9: (page 344) 1, 2, 5, 8 (solutions)

Notes:

• lecture 22

Lecture 23: Systems of ODEs III: Repeated Eigenvalues

Homework:

• Braun, Section 3.10: (page 352) 1,2,3,4,5,6,7

Notes:

• lecture 23

Sample Problems for Midterm Exam 3:

problems (note: the last sample problem has been modified from the version that was posted on Friday.)

solutions

Lecture 26: Fourier Series

Notes:

• lecture 26

Homework:

• Braun, Section 5.4: (page 492) 1,2,6,7,9,10,12 (solutions)

Lecture 27: Return to the Heat Equation

Notes:

• lecture 27

Homework:

• Braun, Section 5.5: (page 498) 1,2,3,4
• Braun, Section 5.6: (page 502) 2,4,5,6 (solutions)

Sample Problems for the Final Exam:

problems

solutions