Differential Equations

Overview

Direction fields, existence and uniqueness of solutions - Numerical methods - Linear equations, models - Solution of linear equations, integrating factors - Complex numbers, roots of unity - Complex exponentials; sinusoidal functions - Linear system response to exponential and sinusoiAutonomous equations; the phase line, stability - Linear vs. nonlinear - Modes and the characteristic polynomial - Good vibrations, damping conditions - Related Mathlet: Damped vibrations - Exponential response formula, spring drive - Related Mathlet: Harmonic frequency response: Variable input frequency - Complex gain, dashpot drive - Operators, undetermined coefficients, resonance - Frequency response - LTI systems, superposition, RLC circuits - Engineering applications

Fourier series - Operations on fourier series - Periodic solutions; resonance - Step functions and delta functions - Step response, impulse response - Convolution - Laplace transform: basic properties - Application to ODEs - Second order equations; completing the squares - The pole diagram - The transfer function and frequency response - Linear systems and matrices - Eigenvalues, eigenvectors - Complex or repeated eigenvalues - Qualitative behavior of linear systems; phase plane - Normal modes and the matrix exponential - Nonlinear systems - Linearization near equilibria; the nonlinear pendulum - Limitations of the linear: limit cycles and chaos