Introduction to Differential Equations

Upon successful completion of the course, students will be able to:

• identify an ordinary differential equation and classify it by order and linearity.
• determine whether or not a unique solution to a first-order initial-value problem exists.
• describe the 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 determine 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 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.
• use the Wronskian to determine whether or not a set of solutions to a differential equation are linearly dependent or independent.
• use reduction of order to find a second solution from a known solution (optional).
• solve homogeneous linear equations with constant coefficients.
• express linear differential equations in terms of differential operators (optional).
• solve nonhomogeneous linear differential equations using the method of undetermined coefficients.
• 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 or 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 (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 their inverse transforms.
• use the convolution theorem and translation theorems to find Laplace transforms and their inverses.
• use the Laplace transform to determine the system transfer function of a linear differential equation.
• use the Laplace transform and the Dirac delta function to determine the impulse response of a linear differential equation.
• 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).
• express higher-order linear differential equations as a first-order system in normal form.
• determine eigenvalues and eigenvectors of a real-valued matrix.
• solve systems of homogeneous first-order linear differential equations using matrix methods.
• solve systems of nonhomogeneous linear first-order differential equations.
• find trajectories associated with, determine critical points of, and perform phase plane analyses for simple autonomous linear and non-linear systems of differential equations.
• model and solve application problems using systems of first-order linear differential equations, including topics such as: electrical circuits, mixtures, chemical reactions, particle dynamics, competition models.
• model and solve application problems using systems of first-order non-linear differential equations, including topics such as the pendulum or predator-prey models.