PHYS 243 – Introduction to Linear Algebra and Differential Equations

Description

This course covers the basics of linear algebra and differential equations, with an emphasis on general physical and engineering applications. It is aimed at students who need the material for future coursework.

Topics

1.  First-Order Differential Equations
   Differential equations and mathematical models
   General and particular solutions
   Linear equations and integrating factors
   Separable equations
   Exact equations
2. Mathematical Models and Numerical Methods
   Population models
   Equilibirum solutions and stability
   Phase portraits
   Euler’s method and improved Euler’s method
3. 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
4. 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
5. 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
6. Eigenvalues and Eigenvectors
   Eigenvalues and eigenvectors of matrices
   Diagonalization of matrices
7. Linear systems of Differential Equations
   First-Order systems and applications
   The Eigenvalue method for linear systems
   Classification of phase portraits of linear planar systems
8. Matrix Exponentials
   Matrix exponentials and linear systems
   Nonhomogeneous linear systems
9. Nonlinear systems
   Stability and the phase plane for nonlinear systems; linear and almost-linear systems
10. Laplace Transform Methods
    Laplace transforms and their inverses
    Applications to piecewise continuous forcing functions

Course descriptions may be subject to occasional minor modifications at the discretion of the instructor.

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