Below are the lecture notes for every lecture session along with links to the Mathlets used during lectures.
| LEC# | TOPICS | RELATED MATHLETS |
|---|---|---|
| I. First-order differential equations | ||
| 1 |
Direction fields, existence and uniqueness of solutions (PDF) |
Related Mathlet: Isoclines |
| 2 |
Numerical methods (PDF) |
Related Mathlet: Euler's method |
| 3 | Linear equations, models (PDF) | |
| 4 | Solution of linear equations, integrating factors (PDF) | |
| 5 | Complex numbers, roots of unity (PDF) | |
| 6 |
Complex exponentials; sinusoidal functions (PDF) |
Related Mathlets: Complex roots, Complex exponential |
| 7 |
Linear system response to exponential and sinusoidal input; gain, phase lag (PDF) |
Related Mathlet: Trigonometric identity |
| 8 |
Autonomous equations; the phase line, stability (PDF) |
Related Mathlet: Phase lines |
| 9 | Linear vs. nonlinear (PDF) | |
| 10 | Exam I | |
| II. Second-order linear equations | ||
| 11 | Modes and the characteristic polynomial (PDF) | |
| 12 |
Good vibrations, damping conditions (PDF) |
Related Mathlet: Damped vibrations |
| 13 |
Exponential response formula, spring drive (PDF) |
Related Mathlet: Harmonic frequency response: Variable input frequency |
| 14 |
Complex gain, dashpot drive (PDF) |
Related Mathlet: Amplitude and phase: Second order II |
| 15 | Operators, undetermined coefficients, resonance (PDF) | |
| 16 |
Frequency response (PDF) |
Related Mathlets: Amplitude and phase: Second order II, Amplitude and phase: First order, Amplitude and phase: Second order III |
| 17 |
LTI systems, superposition, RLC circuits (PDF) |
Related Mathlet: Series RLC circuit |
| 18 |
Engineering applications (PDF) |
|
| 19 | Exam II | |
| III. Fourier series | ||
| 20 |
Fourier series (PDF) |
Related Mathlet: Fourier coefficients |
| 21 |
Operations on fourier series (PDF) |
Related Mathlet: Fourier coefficients: Complex with sound |
| 22 | Periodic solutions; resonance (PDF) | |
| 23 | Step functions and delta functions (PDF) | |
| 24 | Step response, impulse response (PDF) | |
| 25 |
Convolution (PDF) |
Related Mathlets: Convolution: Accumulation, Convolution: Flip and drag |
| 26 | Laplace transform: basic properties (PDF) | |
| 27 | Application to ODEs (PDF) | |
| 28 | Second order equations; completing the squares (PDF) | |
| 29 |
The pole diagram (PDF) |
Related Mathlets: Amplitude response: Pole diagram, Poles and vibrations |
| 30 | The transfer function and frequency response (PDF) | |
| 31 | Exam III | |
| IV. First order systems | ||
| 32 | Linear systems and matrices (PDF) | |
| 33 |
Eigenvalues, eigenvectors (PDF) |
Related Mathlets: Linear phase portrait: Matrix entry, Matrix vector |
| 34 |
Complex or repeated eigenvalues (PDF) |
Related Mathlet: Linear phase portrait: Matrix entry |
| 35 |
Qualitative behavior of linear systems; phase plane (PDF) |
Related Mathlets: Linear phase portrait: Matrix entry, Linear phase portrait: Cursor entry |
| 36 | Normal modes and the matrix exponential (PDF) | |
| 37 | Nonlinear systems (PDF) | |
| 38 | Linearization near equilibria; the nonlinear pendulum (PDF) | |
| 39 |
Limitations of the linear: limit cycles and chaos (PDF) |
Related Mathlet: Vector fields |
| 41 | Final exam | |