TOPICS TO BE COVERED
1. First order equations.
Separable and Linear equations; Direction fields. (§2.1, §2.3)
2. Linear second order differential equations.
Constant coefficient equations (§3.1, §3.4 — with a digression on complex numbers)
Theory for linear equations: Superposition principle, fundamental sets
of solutions (§3.2), the Wronskian test (§3.3).
Reduction of Order (§3.5).
Inhomogeneous equations, the method of undetermined coefficients (§3.6).
3. Linear higher order differential equations.
Homogeneous equations with constant coefficients (§4.2)
Method of undetermined coefficients for inhomogeneous equations(§4.3).
4. Series solutions.
Review of power series and Taylor's formula (§5.1)
Series solution at an “Ordinary Point” (§5.2)
Series solutions at a “Regular Singular Point” (§5.4, §5.5, §5.6)
5. The Laplace Transform.
Definition and basic properties (§6.1)
Using “L ” to solve initial value problems (i.e. the basic use of the Laplace transform—§6.2)
Step functions and discontinuous forcing terms (§6.3, §6.5)
The Convolution Integral (§6.6)
6. Systems of linear equations with constant coefficients.
I will deviate from the presentation in the text book! and show you how you can use the Laplace transform to solve these systems. I think this is a simplification over the book's presentation, but it does presume you can use the Laplace transform. The following lists the sections from which I will assign homework:
Homogeneous linear systems (§7.5)
Coping with Complex Eigenvalues (§7.6 — since we're using theLaplace transform we won't really notice that complex eigenvaluesare a problem.)
Repeated Eigenvalues (§7.7)
Nonhomogeneous systems (§7.9)
7. Nonlinear Systems and Stability.
Depending on how much time is left I will do as many of the following topics as possible:
The phase plane and linear systems in the phase plane (§9.1)
Autonomous systems and stability (§9.2)
Almost linear systems (§9.3)
“Competing Species” (§9.4) and “Predator–Prey” (§9.5) equations.