Differential Equations and Linear Algebra

Contents: Linear Algebra: Matrices, Vectors: Addition and Scalar Multiplication, Matrix Multiplication, Linear Systems of Equations, Gauss Elimination, Linear Independence, Rank of a Matrix, Vector Space, Solutions of Linear Systems: Existence, Uniqueness, Determinants, Cramer's Rule, Inverse of a Matrix, Gauss-Jordan Elimination, Vector Spaces, Inner Product Spaces. Linear Transformations, Matrix Eigenvalue Problem, Determining Eigenvalues and Eigenvectors, Some Applications of Eigenvalue Problems. Ordinary Differential Equations: First-Order ODEs: Basic Concepts, Modeling, Euler's Method, Separable ODEs, Exact ODEs, Integrating Factors, Linear ODEs, Bernoulli Equation, Orthogonal Trajectories, Existence and Uniqueness of Solutions for Initial Value Problems. Second-Order Linear ODEs: Homogeneous Linear ODEs of Second Order, Homogeneous Linear ODEs with Constant Coefficients, Modeling of Free Oscillations of a Mass-Spring System, Euler-Cauchy Equations, Existence and Uniqueness of Solutions, Wronskian, Nonhomogeneous ODEs, Solution by Variation of Parameters.