Differential equations are an important branch of mathematics. They have a rich mathematical formalization, as well as a very successful history of being applied to important problems in physics, chemistry, engineering, and biology. This course will introduce primarily linear, first and second order differential equations. Solution techniques for separable equations, homogeneous and inhomogeneous equations, as well as an intuition for modeling-based applications will be presented. The application of Laplace transforms to differential equations, systems of linear differential equations, linearization of nonlinear systems, and phase plane methods will be introduced. Fourier series and their application to simple partial differential equations will be treated. MATLAB based numerical solution and visualization will be briefly covered.
On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.
Week 1: August 26, 28, 30
Modeling with differential equations
First order initial value problems
Separation of variables
1.1 Number 1-11. Homework: 1,2,5,7,11
2.1 Number 1-6, 12-15. Homework: 1,3,5,12,13,15
2.2 Number 1-18, 33-35 Homework: 3,5,9,33
M1 Number 1
Solutions
Week 2: September 4, 6
Models of motion
First order linear equations
Plotting with MatLab
2.3 Number 8-10 Homework: 9
2.4 Number 1-21 Homework: 5,15,19
M2 Number 15-20
Solutions
Week 3: September 10, 12, 14
Modeling
dfield
3.1 Number 9, 11, 13 Homework: 10,13
3.3 Number 3, 5 Homework: 3,5
M3 Number 1-12 Homework: 1
Solutions
Week 4: September 16, 18, 20
Electrical circuits
Second order equations
Function m-files
3.4 Number 1-10 Homework: 1,3,5,7,11
4.1 Number 1-20 Homework: 1,3,9,17
4.3 Number 1-36 Homework: 1,9,17,35
M4 Number 1-8, 17, 18
Solutions
Week 5: September 23, 25, 27
Second order equations (cont.)
4.4 Number 1-12 Homework: 1,7,11,12
4.5 Number 1-29 Homework: 1,5,11,15,19
4.6 Number 1-10 Homework: 1,3,5
Solutions
Week 6: October 1, 2, 4
Forced Harmonic Motion
Review and Exam
4.7 Number 3-6, 12-15 Homework: 3,13
Solutions
First midterm on October 4
Week 7: October 7, 9, 11
Laplace Transform
5.1 Number 1-29 Homework: 7,13,15,29
5.2 Number 1-41 Homework: 5,11,19,29
5.3 Number 1-36 Homework: 3,7,11,19
Solutions
Week 8: October 14, 16, 18
Laplace Transform (cont.)
5.4 Number 1-26 Homework: 7,11,21
5.5 Number 1-25 Homework: 1,3,11,17
5.6 Number 1-9 Homework: 2,3,5,7
Solutions
Week 9: October 21, 23, 25
Convolutions
Numerical methods
Introduction to systems
5.7 Number 4-10 Homework: 6, 8, 10
6.1 Number 1-5 Homework: 3,5
8.1 Number 1-16 Homework: 5,7,13,15
M5 Number 1-6 Homework: 2,3
Solutions
Week 10: October 28, 30, November 1
Systems (cont.)
Constant coefficient homogeneous 2x2 systems
8.2 Number 13-16 (use pplane 8) Homework: 11,13,15
8.3 Number 1-6 Homework: 1,3,5
9.1 Number 1-8, 16-23 Homework: 3,5,17,19
Solutions
Week 11: November 4, 6, 8
Planar systems
Review and Exam
9.2 Number 1-27, 58, 59 Homework: 3,13,15,41,49
Solutions
Second Midterm on November 8
Week 12: November 11, 13, 15
Phase plane
Inhomogeneous systems: Undetermined coefficients and variation of parameters
Nonlinear systems: equilibria, linearization
9.3 Number 1-23 Homework: 1,11, 13, 15, 17, 21, 23
9.9 Number 1-6, 12-15 Homework: 1, 3, 13, 15
10.1 Number 1-18 Homework: 3, 5, 7, 15
Solutions
Week 13: November 18, 20, 22
Nonlinear systems: stability, nullclines
10.2 Number 1-4 Homework: 1,3
10.3 Number 1-16 Homework: 3,7,11
12.1 Number 1-17 Homework: 5,7,13,17
Solutions
Week 14: November 25
Nonlinear systems: invariant sets, nullclines
Fourier series
12.3 Number 1-32 Homework: 3,7,19,31
Solutions
Week 15: December 2, 4, 6
Heat equation
Separation of variables
Review
Review
13.1 Number 1-9
13.2 Number 1-18 Homework: 5, 13