Students will be able to derive the governing differential equations (DEs) that describe familiar physical processes that arise in science and engineering. The methods of derivation involve linearization, compartmental analysis, Newton’s laws, financial mathematics, conservation of energy, and Kirchoff’s law.
Students will learn solution techniques for first order, separable, and linear DEs; and be able to solve initial value problems.
Students will understand how to find approximate numerical solutions and develop the ability to use software such as Matlab, Python, or internet-based tools as appropriate to solve for and interpret DE results.
Students will be able to visualize solution graphs and numerical approximations to initial value problems via slope fields.
Students will become fluent in matrix algebra techniques in order to be able to compute the solution space to linear systems and understand its structure.
Students will be able to use the concepts of linear vector spaces such as linear combinations, span, independence, basis, and dimension, to understand the solution space to linear equations, linear DEs, and linear systems of DEs.
Students will learn how to solve constant coefficient linear DEs via superposition to find particular solutions to homogeneous and non-homognenous problems via characteristic equation analysis.
Students will be able to find eigenvalues and eigenvectors of matrices, and use them to find the solution space to first and second order constant coefficient homogeneous linear systems of DEs.
Students will understand and be able to use linearization as a technique to understand the behavior of nonlinear autonomous dynamical systems near equilibrium solutions.
Students will learn how to use Laplace transform techniques to solve linear differential equations involving discontinuous forcing using integral convolutions.
In addition to topical content, students will learn to understand problem descriptions, select the appropriate operations, execute methods accurately, and finally interpret and communicate results.