A Second Course in Differential Equations

Delve into advanced topics in ordinary differential equations through this comprehensive 5-hour course. Explore linear systems, solution set independence, eigenvalues, and eigenvectors. Master the matrix format of linear differential equation systems and tackle example problems involving the Wronskian. Study repeated real eigenvalues, complex eigenvalues, and notation changes. Learn to solve nonhomogeneous linear systems using undetermined coefficients and variation of parameters methods. Discover numerical methods, including the Euler method and its improvements, as well as the fourth-order Runge-Kutta method. Gain practical skills in using Python for numerical calculations, from installation to constructing code for various formulas. Apply these techniques to solve second-order ODEs, systems of first-order ODEs, and higher-order ODEs. Conclude with an exploration of plane autonomous systems and their applications through example problems.

A01 Introduction to linear systems.
A02 Independence of the solution set.
A03 Eigenvalues.
A04 Eigenvectors.
A05 Explanation of the matrix format of a system of linear differential equations.
A06 Example problem including the Wronskian.
A07 Repeated real eigenvalues.
A08 Example problem of repeated real eigenvalues.
A09 Example problem of multiplicity two.
A10 Example problem of multiplicity three.
A11 Eigenvalues with complex numbers.
A12 Changing the notation.
A13 More Example Problems.
A14 Nonhomegeneous linear systems solved by undetermined coefficients.
A15 More about undetermined coefficients.
A16 The method of variation of parameters.
A17 Deriving the equation for the particular solution.
A18 Example problem using variation of parameters.
B01 An introduction to numerical methods.
B02 An introduction to the Euler method.
B03 An improvement of the Euler method.
B04 An example problem.
B05 Local truncation errors.
B06 Example problem calculating the error.
B07 Fourth-order Runge Kutta.
B08 Using Python to do numerical calculations.
B09 Installing Python on the Mac.
B10 Constructing code for the Euler Formula.
B11 The improved Euler Formula.
B12 Second order ODEs solved with Python using the Euler formula.
B13 Second order ODE solved with RK4 in Python.
B14 Simplifying a system of higher order ODEs.
B15 Solving a system of first order ODEs with RK4 using Python.
B16 Example problem solving a higher order ODE with the Euler formula.
B17 Example problem solving a system of ODEs with RK4.
B18 Plane autonomous systems.
B19 Example problem of a system of autonomous equations.
B21 Example problem.