MAT275 - Modern Differential Equations
This course covers basics of ordinary differential equations (ODEs) from an
applied point of view. This course provides a streamlined exposition of
methods for first-order ODEs and systems of ODEs and focuses on practical
aspects of numerical techniques for solving a wide range of applied problems,
using Matlab. The course includes an extended Matlab project.
Topics
- syllabus,
- calculus review,
- introduction to Matlab
- integration review,
- Lab 1: direction fields ,
- Lab 1: direction fields (new version) ,
- scalar ODEs with Matlab,
- Matlab example with ode45.m
- Models
- scalar: population, Newton law of cooling, mixing, chemical reaction
- 2nd order scalar: spring-mass system, electrical circuit
- systems: SIR model, predator-prey systems, discretized heat equation
- Lab 2: Euler/improved
Euler for scalar ODEs
- Lab 5: models,
Matlab example with euler.m
- separable scalar ODEs, homogeneous scalar linear ODEs
- systems of ODEs,
systems of ODEs
with Matlab,
example with 2 ODEs, reduction
- discussion of project
(identical for all students, especially in summer sessions or see the
group projects setting during a regular semester).
- scalar linear forced ODEs, variation of constant
- second-order linear HODEs: superposition principle
- characteristic equation, second-order linear HODEs: exponential solutions
- Lab 8: spring-mass system
- forced second-order linear ODEs: undetermined coefficients
- Lab 10: forced spring-mass
system (resonance, beats)
- linear 2x2 systems, matrix formulation, eigenvalues
- Lab 11: phase plot
for 2x2 linear systems
- solution of 2x2 HODE system
- Lab 7: ode45, Euler for
systems of ODEs, non-homogeneous linear 2x2 ODE systems
(variation of constants)
- Laplace transform
- Laplace transform methods for linear scalar and systems of ODE
- Lab 3: existence,
uniqueness of solutions
- linearization of autonomous systems, stability of equilibrium
- Lab 13: Simulink
- series solution
Resources
- Homework assignments
- ex1.m: Matlab example with ode45.m
- ex2.m: Matlab example with euler.m
- ex3.m: Matlab example with 2 ODEs/ode45.m,
- project
- calculus review (trig functions,...)
- integration review (partial fractions,...) (updated 6/02/05),
key to problems
- complete Matlab manual (Labs) (2.13Mb), or
- Matlab m-files and
Simulink models .mdl files used in the above manual
-
most common errors in undergraduate math (by Eric Schechter)
-
common math errors (by Paul Cox)
- ode book (by Paul Cox, 1.94Mb)
- old tests/quizzes from
MAT274/MAT275
- John Polking's
Matlab manual (including a guide to dfield6)
- The best Matlab/Simulink reference is the original source at
The MathWorks
(look in particular in the "Documentation" menu).
- additional basic online tutorials
tutorial 1,
tutorial 2,
tutorial 3,
tutorial 4,
tutorial 5
- Matlab FAQ
- miscellaneous files and links