Introduction to Differential Equations

Upon completion of MATH 2421 the student should be able to:

• identify an ordinary differential equation and classify it by order or linearity
• determine whether or not a unique solution to a first-order initial-value problem exists
• understand differences between solutions of linear and non-linear first-order differential equations
• recognize and solve linear, separable and exact first-order differential equations
• use substitutions to solve various first-order differential equations (optional)
• recognize and solve autonomous first-order differential equations, analyze trajectories, and comment on the stability of critical points
• use the Euler method to approximate solutions to first-order differential equations
• model and solve application problems using linear and non-linear first-order differential equations, including, but not limited to, topics such as: growth and decay, series circuits, Newton’s Law of Cooling, mixtures, logistic growth, chemical reactions, particle dynamics
• determine whether or not a unique solution to a linear nth-order initial-value problem exists
• determine whether or not a set of solutions to a differential equation are linearly dependent or independent using the Wronskian
• use reduction of order to find a second solution from a known solution
• solve homogeneous linear equations with constant coefficients
• express linear differential equations in terms of differential operators (optional)
• use the method of undetermined coefficients to solve nonhomogeneous linear differential equations for which the nonhomogeneous term can be annihilated
• solve nonhomogeneous linear differential equations using variation of parameters
• solve nonhomogeneous linear differential equations using Green’s functions (optional)
• model, solve and analyze problems involving mechanical and electrical vibrations using second-order linear differential equations
• determine ordinary and singular points of linear differential equations
• recognise and solve Cauchy-Euler equations
• use power series techniques to solve linear differential equations in the neighbourhood of ordinary points
• use the method of Frobenius to solve linear differential equations about regular singular points
• use series methods to solve Bessel, modified Bessel and Legendre equations (optional)
• state the definition of the Laplace transform of a function and the sufficient conditions for its existence
• determine the Laplace transforms for basic functions, derivatives, integrals and periodic functions and find inverse transforms
• use the convolution theorem and translation theorems to find Laplace transforms and their inverses
• use Laplace transforms to solve initial value problems, integral equations and integro-differential equations
• solve systems of differential equations using differential operators or Laplace transforms (optional)
• reduce higher-order linear differential equations to first-order systems in normal form
• solve systems of homogeneous first-order linear differential equations using matrix methods
• solve systems of nonhomogeneous linear first-order differential equations
• model and solve application problems using systems of first-order linear differential equations, including, but not limited to, topics such as: parallel circuits, mixtures, chemical reactions, particle dynamics, competition models
• find trajectories associated with, determine critical points of, and perform phase plane analyses for simple autonomous linear and non-linear systems of equations