**Differential Equations and Linear Algebra** |

**Introduction** |

Lecture 01 - Overview of Differential Equations |

Lecture 02 - The Calculus You Need |

**First Order Equations** |

Lecture 03 - Response to Exponential Input |

Lecture 04 - Response to Oscillating Input |

Lecture 05 - Solution for Any Input |

Lecture 06 - Step Function and Delta Function |

Lecture 07 - Response to Complex Exponential |

Lecture 08 - Integrating Factor for Constant Rate |

Lecture 09 - Integrating Factor for a Varying Rate |

Lecture 10 - The Logistic Equation |

Lecture 11 - The Stability and Instability of Steady States |

Lecture 12 - Separable Equations |

**Second Order Equations** |

Lecture 13 - Second Order Equations |

Lecture 14 - Forced Harmonic Motion |

Lecture 15 - Unforced Damped Motion |

Lecture 16 - Impulse Response and Step Response |

Lecture 17 - Exponential Response ? Possible Resonance |

Lecture 18 - Second Order Equations with Damping |

Lecture 19 - Electrical Networks: Voltages and Currents |

Lecture 20 - Method of Undetermined Coefficients |

Lecture 21 - An Example of Undetermined Coefficients |

Lecture 22 - Variation of Parameters |

Lecture 23 - Laplace Transform: First Order Equation |

Lecture 24 - Laplace Transform: Second Order Equation |

Lecture 25 - Laplace Transforms and Convolution |

**Graphical and Numerical Methods** |

Lecture 26 - Pictures of Solutions |

Lecture 27 - Phase Plane Pictures: Source, Sink, Saddle |

Lecture 28 - Phase Plane Pictures: Spirals and Centers |

Lecture 29 - Two First Order Equations: Stability |

Lecture 30 - Linearization at Critical Points |

Lecture 31 - Linearization of Two Nonlinear Equations |

Lecture 32 - Eigenvalues and Stability: 2 by 2 Matrix, A |

Lecture 33 - The Tumbling Box in 3-D |

**Vector Spaces and Subspaces** |

Lecture 34 - The Column Space of a Matrix |

Lecture 35 - Independence, Basis, and Dimension |

Lecture 36 - The Big Picture of Linear Algebra |

Lecture 37 - Graphs |

Lecture 38 - Incidence Matrices of Graphs |

**Eigenvalues and Eigenvectors** |

Lecture 39 - Eigenvalues and Eigenvectors |

Lecture 40 - Diagonalizing a Matrix |

Lecture 41 - Powers of Matrices and Markov Matrices |

Lecture 42 - Solving Linear Systems |

Lecture 43 - The Matrix Exponential |

Lecture 44 - Similar Matrices |

Lecture 45 - Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors |

Lecture 46 - Second Order Systems |

**Applied Mathematics and ATA** |

Lecture 47 - Positive Definite Matrices |

Lecture 48 - Singular Value Decomposition (the SVD) |

Lecture 49 - Boundary Conditions Replace Initial Conditions |

Lecture 50 - Laplace Equation |

**Fourier and Laplace Transforms** |

Lecture 51 - Fourier Series |

Lecture 52 - Examples of Fourier Series |

Lecture 53 - Fourier Series Solution of Laplace's Equation |

Lecture 54 - Heat Equation |

Lecture 55 - Wave Equation |