Math 308-507, Fall 2016, Tentative Weekly Schedule
08/29 Solutions of some differential equations (first order linear equations
with constant coefficients, section 1.2) Separable equations
(section 2.2)
08/31 Separable equations (section 2.2 continued), direction field (sections 1.1), autonomous equations, and population dynamics (sections 1.1 and 2.5
combined);
09/05 Autonomous equations and population dynamics (section 2.5
continued). Linear equations: Method of integrating factors (section 2.1);
09/07 Linear equations: Method of integrating factors (section 2.1 continued). Modelling with first order equations (section 2.3);
09/12 Differences between linear and non-linear equations (section 2.4). Exact equations and integrating factors (section 2.6 continued). (section 2.6);
09/14 Solutions of linear homogeneous equations of second order, the Wronskian (section 3.2); Linear
homogeneous equations of second order with constant coefficients: the
cases of distinct roots and repeated roots of characteristic equation
(sections 3.1 and 3.4)
09/19 Review of complex numbers. Linear homogeneous equations of second order with constant coefficients: the case of complex roots of the characteristic equation (section 3.3);
09/21 Mechanical and electrical vibrations (section 3.7);
Introduction and review of matrices (section 7.1, 7.2)
09/26 Linear algebraic equations; linear independence (section 7.3).
Basic theory of systems of first order linear equations (section 7.4);
09/28 Eigenvalues, eigenvectors (section 7.3);Homogeneous linear systems with constant coefficients, the case of distinct eigenvalues (section 7.5);
10/03 Complex eigenvalues (section 7.6);
10/05 Fundamental matrices , matrix exponential (section 7.7); Repeated eigenvalues:
algebraic and geometric multiplicity, the case when geometric
multiplicity is equal to algebraic (part of section 7.5).
10/10 Repeated eigenvalues: the
case when geometric multiplicity is smaller than the algebraic
multiplicity for some eigenvalues, the notion of generalized
eigenvectors and calculation of the exponential of the matrix of the
system in a basis of generalized eigenvectors (extended material of
section 7.8 and my own lecture notes).
10/12 Repeated
eigenvalues: the case of n=2 and n=3, all possible cases of geometric
and algebraic multiplicities for these dimensions (material of section
7.8 based on my own notes)
10/17 The Phase Plane; Linear Systems, classification of critical points (section 9.1);
10/19 The phase plane; linear
system: classification of mcritical points (continued). Autonomous
Systems, Critical points, and Stability (section 9.2); Locally linear systems (section 9.3)
10/24 Locally linear systems (section 9.3, continued). Examples: competing species (section 9.4)
10/26
Non-homogeneous linear equations and linear systems of equations .
Method of undetermined coefficients (section 3.5 and section 7.9
combined );
10/31 Method of variation of parameters both for scalar linear equations and
linear systems of equations (section 3.6 and section 7.9 combined);
11/07 Definition of Laplace transform (section 6.1). Solution of initial value problems using Laplace transform (section 6.2);
11/09 Solution
of initial value problems using Laplace transform (section 6.2 continued and part of section 7.9 concerning Laplace transform).
11/14 Step functions.
Differential equations with discontinuous forcing functions (sections
6.3 and 6.4);
11/16 Impulse functions (section 6.5);
The convolution integral; 11/21 Review of power series (section 5.1); Series solutions near an ordinary point, part I (section 5.2);
11/28 Series solutions near an ordinary point, part II (section 5.3);
Euler equations: Regular singular points (section 5.4)
11/30 Series solution near a regular singular point, part I (section 5.5);
12/07. Series solution near a regular singular point, part II (section 5.6); Review before the final exam.