Differential Equations

Lecturer: Professor Arthur Mattuck
College/University: Massachusetts Institute of Technology (MIT)

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: Behavior of Solutions, Use of Complex Methods
   
8. Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models
   
9. Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases
   
10. Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations
    
11. Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians
    
12. Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's
    
13. Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Exponentials
    
14. Interpretation of the Exceptional Case: Resonance
    
15. Introduction to Fourier Series; Basic Formulas for Period 2(pi)
    
16. Continuation: More General Periods; Even and Odd Functions; Periodic Extension
    
17. Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds
    
18. Introduction to the Laplace Transform; Basic Formulas
    
19. Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's
    
20. Using Laplace Transform to Solve ODE's with Discontinuous Inputs
    
21. Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
    
22. Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System
    
23. Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients
    
24. Matrix Exponentials; Application to Solving Systems
    
25. Decoupling Linear Systems with Constant Coefficients
    
26. Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum
    
27. Limit Cycles: Existence and Non-existence Criteria