Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's

Lecture Description

Course Description

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time.

Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

Course Index

1. The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
2. Euler's Numerical Method for y'=f(x,y) and its Generalizations
3. Solving First-order Linear ODE's; Steady-state and Transient Solutions
4. First-order Substitution Methods: Bernouilli and Homogeneous ODE's
5. First-order Autonomous ODE's: Qualitative Methods, Applications
6. Complex Numbers and Complex Exponentials
7. First-Order Linear with Constant Coefficients
8. Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models
9. Solving Second-Order Linear ODE's with Constant Coefficients
10. Complex Characteristic Roots; Undamped and Damped Oscillations
11. Second-Order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
12. Inhomogeneous ODE's; Stability Criteria for Constant-Coefficient ODE's
13. Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials
14. Interpretation of the Exceptional Case: Resonance
15. Introduction to Fourier Series; Basic Formulas for Period 2(pi)
16. More General Periods; Even and Odd Functions; Periodic Extension
17. Finding Particular Solutions via Fourier Series; Resonant Terms
18. Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
19. Convolution Formula: Proof, Connection with Laplace Transform, Application
20. Using Laplace Transform to Solve ODE's with Discontinuous Inputs
21. Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
22. First-Order Systems of ODE's; Solution by Elimination, Geometric Interpretation
23. Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues
24. Continuation: Repeated Real Eigenvalues, Complex Eigenvalues
25. Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
26. Matrix Methods for Inhomogeneous Systems
27. Matrix Exponentials; Application to Solving Systems
28. Decoupling Linear Systems with Constant Coefficients
29. Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories
30. Limit Cycles: Existence and Non-existence Criteria
31. Non-Linear Systems and First-Order ODE's