Description
The basics of linear algebra and differential equations, with an emphasis on general physical and
engineering applications. Aimed at students who need the material for future coursework but do not
need as much depth and rigor as provided by MATH240/MATH461 and MATH246.
Prerequisites
a C- or better in MATH 141
Topics
- First-Order Differential Equations
Differential equations and mathematical models
General and particular solutions
Linear equations and integrating factors
Separable equations
Exact equations - Mathematical Models and Numerical Methods
Population models
Equilibirum solutions and stability
Phase portraits
Euler's method and improved Euler's method - Linear Systems and Matrices
Matrices and linear systems; row reduction and Gaussian elimination
Reduced row echelon form; matrix operations
Matrix inverses and determinants
Linear equations and curve-fitting - Vector Spaces
R^n as a vector space and subspaces
Linear combinations and independence of vectors
Bases and dimension; column space and row space of a matrix
Orthogonal vectors - Higher-order Differential Equations
Second-order linear equations; principle of superposition
Homogeneous constant-coefficient equations
Non-homogeneous equations; the method of undetermined coefficients
Mechanical vibrations; forced oscillations and resonance - Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of matrices
Diagonalization of matrices - Linear systems of Differential Equations
First-Order systems and applications
The Eigenvalue method for linear systems
Classification of phase portraits of linear planar systems - Matrix Exponentials
Matrix exponentials and linear systems
Nonhomogeneous linear systems - Nonlinear systems
Stability and the phase plane for nonlinear systems; linear and almost-linear systems - Laplace Transform Methods
Laplace transforms and their inverses
Applications to piecewise continuous forcing functions