Differential Equations and Linear Algebra

Preface; 1: First Order Equations; 1.1: Four Examples: Linear versus Nonlinear; 1.2: The Calculus you Need; 1.3: The Exponentials; 1.4: Four Particular Solutions; 1.5: Real and Complex Sinusoids; 1.6: Models of Growth and Decay; 1.7: The Logistic Equation; 1.8: Separable Equations and Exact Equations; 2: Second Order Equations; 2.1: Second Derivatives in Science and Engineering; 2.2: Key Facts About Complex Numbers; 2.3: Constant Coefficients A, B, C; 2.4: Forced Oscillations and Exponential Response; 2.5: Electrical Networks and Mechanical Systems; 2.6: Solutions to Second Order Equations; 2.7: Laplace Transforms Y(s) and F(s); 3: Graphical and Numerical Methods; 3.1: Nonlinear Equations; 3.2: Sources, Sinks, Saddles, and Spirals; 3.3: Linearization and Stability in 2D and 3D; 3.4: The Basic Euler Methods; 3.5: Higher Accuracy with Runge-Kutta; 4: Linear Equations and Inverse Matrices; 4.1: Two Pictures of Linear Equations; 4.2: Solving Linear Equations by Elimination; 4.3: Matrix Multiplication; 4.4: Inverse Matrices; 4.5: Symmetric Matrices and Orthogonal Matrices; 5: Vector Spaces and Subspaces; 5.1: The Column Space of a Matrix; 5.2: The Nullspace of A: Solving Av=0; 5.3: The Complete Solution to Av=b; 5.4: Independence, Basis and Dimension; 5.5: The Four Fundamental Subspaces; 5.6: Graphs and Networks; 6: Eigenvalues and Eigenvectors; 6.1: Introduction to Eigenvalues; 6.2: Diagonalizing a Matrix; 6.3: Linear Systems y =Ay; 6.4: The Exponential of a Matrix; 6.5: Second Order Systems and Symmetric Matrices; 7: Applied Mathematics and ATA; 7.1: Least Squares and Projections; 7.2: Positive Definite Matrices and the SVD; 7.3: Boundary Conditions Replace Initial Conditions; 7.4: Laplace's Equation; 7.5: Networks and the Graph Laplacian; 8: Fourier and Laplace Transforms; 8.1: Fourier Series; 8.2: The Fast Fourier Transform; 8.3: The Heat Equation; 8.4: The Wave Equation; 8.5: The Laplace Transform; 8.6: Convolution (Fourier and Laplace); Matrix Factorizations; Properties of Determinants; Index; Linear Algebra in a Nutshell.