SES # | TOPICS | READINGS AND RELATED MATHLETS |
---|---|---|

I. First-order differential equations | ||

R1 | Natural growth, separable equations | [EP]: 1.1 and 1.4 |

L1 | Direction fields, existence and uniqueness of solutions | [EP]: 1.2 and 1.3 |

L2 | Numerical methods | [EP]: 6.1 and 6.2 [Notes]: G.2 (PDF) Euler's method Mathlet |

L3 | Linear equations, models | [EP]: 1.5 [SN]: 2 (PDF) |

L4 | Solution of linear equations, integrating factors | [EP]: 1.5 [SN]: 3 (PDF) |

L5 | Complex numbers, roots of unity | [SN]: 5 (PDF) [SN]: 6 (PDF) [Notes]: C.1-3 (PDF) |

L6 | Complex exponentials; sinusoidal functions | [SN]: 4 (PDF) [Notes]: C.4 (PDF) and IR.6 (PDF) Complex roots Mathlet Complex exponential Mathlet |

L7 | Linear system response to exponential and sinusoidal input; gain, phase lag | [SN]: 4 (PDF) [Notes]: IR.6 (PDF) Trigonometric identity Mathlet |

L8 | Autonomous equations; the phase line, stability | [EP]: 1.7 and 7.1 [SN]: Appendix A (PDF) Phase lines Mathlet |

L9 | Linear vs. nonlinear | [SN]: Appendix C (PDF) |

II. Second-order linear equations | ||

L11 | Modes and the characteristic polynomial | [EP]: 2.1, 2.2, and 2.3 up to "Polynomial Operators" [SN]: 9 (PDF) |

L12 | Good vibrations, damping conditions | [EP]: 2.3 and 2.4 Damped vibrations Mathlet |

L13 | Exponential response formula, spring drive | [EP]: 2.6, pp. 157-159 [SN]: 7 (PDF) (for beats) [Notes]: O.1 (PDF) Harmonic frequency response: Variable input frequency Mathlet |

L14 | Complex gain, dashpot drive | [EP]: 2.6, pp. 165-167 [SN]: 10 (PDF) [Notes]: O.1, 2, 4 (PDF) Amplitude and phase: Second order II Mathlet |

L15 | Operators, undetermined coefficients, resonance | [EP]: 2.5, pp. 144-153 and EP: 2.7 |

L16 | Frequency response | [SN]: 14 (PDF) Amplitude and phase: Second order II Mathlet Amplitude and phase: First order Mathlet Amplitude and phase: Second order III Mathlet |

L17 | LTI systems, superposition, RLC circuits. | [SN]: Appendix B (PDF) Series RLC circuit Mathlet |

L18 | Engineering applications | [SN]: 12 (PDF) [SN]: 13 (PDF) [Notes]: O.3 (PDF) |

III. Fourier series | ||

L20 | Fourier series | [EP]: 8.1 [SN]: 16 (PDF) Fourier coefficients Mathlet |

L21 | Operations on fourier series | [EP]: 8.2 and 8.3 Fourier coefficients: Complex with sound Mathlet |

L22 | Periodic solutions; resonance | [EP]: 8.3 and 8.4 |

L23 | Step function and delta function | [SN]: 17 (PDF) |

L24 | Step response, impulse response | [SN]: 18 (PDF) [Notes]: IR (PDF) |

L25 | Convolution | [SN]: 18 (PDF) Convolution: Accumulation Mathlet Convolution: Flip and drag Mathlet |

L26 | Laplace transform: basic properties | [EP]: 4.1 |

L27 | Application to ODEs | [EP]: 4.2 and 4.3 [SN]: 20 (PDF) [Notes]: H |

L28 | Second order equations; completing the squares | [EP]: 4.5 and 4.6 [SN]: 20 (PDF) |

L29 | The pole diagram | [EP]: 4.4 [SN]: 22 (PDF) [SN]: 23 (PDF) 18.03 Difference Equations and Z-Transforms (PDF)(Courtesy of Jeremy Orloff.) Amplitude response: Pole diagram Mathlet Poles and vibrations Mathlet |

L30 | The Transfer function and frequency response | |

IV. First order systems | ||

L32 | Linear systems and matrices | [EP]: 5.1-5.3 [SN]: 25 (PDF) [Notes]: LS.1 (PDF) |

L33 | Eigenvalues, eigenvectors | [EP]: 5.4 [Notes]: LS.2 (PDF) Linear phase portrait: Matrix entry Mathlet Matrix vector Mathlet |

L34 | Complex or repeated eigenvalues | [EP]: 5.4 [Notes]: LS.3 (PDF) Linear phase portrait: Matrix entry Mathlet |

L35 | Qualitative behavior of linear systems; phase plane | [SN]: 26 (PDF) Linear phase portrait: Matrix entry Mathlet Linear phase portrait: Cursor entry Mathlet |

L36 | Normal modes and the matrix exponential | [EP]: 5.7 [Notes]: LS.6 (PDF) |

L37 | Nonlinear systems | [EP]: 7.2 and 7.3 [Notes]: LS.6 (PDF) |

L38 | Linearization near equilibria; the nonlinear pendulum | [EP]: 7.4 and 7.5 [Notes]: GS (PDF) [SN]: Appendix B (PDF) [SN]: Appendix C (PDF) |

L39 | Limitations of the linear: limit cycles and chaos | Vector fields Mathlet |