San Jose State University
Department of Mathematics

MATH 133A
Ordinary Differential Equations


Catalog Description:

First order differential equations, first order linear systems, second order linear equations, applications, Laplace transforms, series solutions. Additional topics. (3 units)

Prerequisites:

Prerequisite: Math 32 (with a grade of "C-" or better) or instructor consent.

Topics:
1. First order differential equations: definition of solution, existence and uniqueness, slope fields, separation of variables, linear ODEs, phase line and equilibria, changing variables. Euler's method.
2. First order (planar) linear systems: modeling examples, geometry of systems. Linear algebra preliminaries (matrices, matrix multiplication, determinants). Straight-line solutions, eigenvectors and eigenvalues, phase plane for linear systems with real distinct/ complex/real repeated eigenvalues, the trace-determinant plane.
3. Linear second order equations: reduction to systems, characteristic equation and general solution of the homogeneous equation, method of undetermined coefficients. Forced mechanical vibrations.
4. Laplace transform: definition, properties, inverse transform, solving initial value problems, transforms of discontinuous functions, convolution, impulses and Dirac delta function.
5. Series solutions: Taylor series method, power series, analytic functions.


Possible textbooks:
Blanchard, Devaney, Hall, Ordinary Differential Equations, Brooks/Cole (Thompson), 3rd edition, 2006
Polking, Boggess, Arnold, Differential Equations, 2nd edition, Pearson/Prentice Hall, 2005

MATH 133A COURSE SCHEDULE

1. First order differential equations. (3 weeks)
– Definition of a solution, existence and uniqueness, slope fields: 1 week
– Separation of variables, linear ODEs: 1 week
– Phase line and equilibria, changing variables 1 week
2. First order (planar) linear systems. (5 weeks)
– Modeling examples, geometry of systems: 1 week
– Linear algebra preliminaries (matrices and matrix multiplication, determinants): 1 week
– Straight-line solutions, eigenvalues and eigenvectors: 1 week
– Phase plane for planar linear systems (real distinct eigenvalues, complex eigen-
values, real repeated eigenvalues), trace-determinant plane: 2 weeks.
3. Linear second order equations. (2 weeks)
– Reduction to systems, characteristic equation and general solution to the homogeneous equation: 1 week
– Method of undetermined coefficients. Forced mechanical vibrations: 1 week
4. Laplace transform. (4 weeks)
– Definition, properties: 1 week
– Inverse transform, solving initial value problems: 1 week
– Transforms of discontinuous functions, convolution: ∼ 1.5 weeks
– Impulses and Dirac delta function: ∼ 1.5 weeks
5. Series solutions. (1 week)
– Taylor series method, power series and analytic functions: 1 week

Last modified by Slobodan Simić on March 25, 2009