MAT3613

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Differential equations: what they are and why they are important; order of a differential equation; solutions; initial value problems. Theory of first order differential equations; Cauchy problem; existence and uniqueness. The method of separation of variables; examples of separable differential equations. HOMEWORK # 1 – First Order ODEs: Due at the beginning of Week IV	Integration techniques.	Explain the basic notions related to differential equations: the order of a differential equation, solutions, basic concepts of initial values, existence and uniqueness. Determine separable differential equations of the first order. Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
Week II	Ahmad and Ambrosetti 2014, Chaps. 1 and 3	Homogeneous differential equations. Linear differential equations of first order; integrating factors.	Integration techniques.	Determine homogeneous and linear differential equations of the first order. Apply direct methods to evaluate exact solutions of homogeneous and linear differential equations of the first order (substitutions, integrating factor method). Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
Week III	Ahmad and Ambrosetti 2014, Ch. 3	Bernoulli equations. Exact differential equations. The integrating factor for exact equations.	Integration techniques. Partial derivatives. Linear first- order differential equations.	Determine Bernoulli and exact differential equations of the first order. Apply direct methods to evaluate exact solutions of Bernoulli and exact differential equations of the first order. Use the integrating factor technique for exact equations.
Week IV	Ahmad and Ambrosetti 2014, Chaps. 1-3	Collect HOMEWORK # 1 Firs-order ODEs not solved for the first derivative: Clairaut equations, Lagrange equations. Overview of the solutions methods discussed so far (Chapters 1-3).	Integration techniques. Partial derivatives. Integrating factor methods. First-order differential equations.	Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact. Use direct methods to solve first order differential equations solved and not solved for the first derivative.
Week V	Ahmad and Ambrosetti 2014, Ch. 5	MDTERM EXAM # 1: First-order ODEs Linear independence and Wronskian.	Linear dependence, independenc e of vectors. Determinant s.	Linear dependence and independence of functions. Wronskian of two functions. Wronskian of two solutions of linear second-order ODEs.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Reduction of the order. Linear homogeneous differential equations. Abel’s theorem. Fundamental solutions. Linear nonhomogeneous equations; variation of parameters. HOMEWORK # 2 – Second and higher order ODEs: Due at the beginning of Week X (extended later)	Wronskian. Algebraic equations. Determinant s.	Determine the type of different classes of differential equations of the second and higher order: linear and nonlinear, equations with constant coefficients, homogeneous and non- homogeneous. Determine fundamental solutions. Apply of the variation of parameters technique for second-order ODEs.
Week VII	Ahmad and Ambrosetti 2014, Ch. 5	Variation of parameters (continued) Method of undetermined coefficients	Variation of parameters. Method of undetermined coefficients.	Apply variation of parameters and method of undetermined coefficients techniques for second-order ODEs.
Week VIII		SPRING BREAK
Week IX		Preparation for remote instruction.
Week X	Ahmad and Ambrosetti 2014, Ch. 5	Higher order ODEs.	Methods for higher-order ODEs. Variation of parameters. Method of undetermined coefficients.	Apply variation of parameters and method of undetermined coefficients techniques for higher-order ODEs
Week XI	Ahmad and Ambrosetti 2014, Chaps. 5, 6, 10	Overview of the solutions methods for second and higher order differential equations. Collect HOMEWORK # 2 (extended deadline)	Direct methods for second and higher-order ODEs.	Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
Week XII	Ahmad and Ambrosetti 2014, Ch. 11	MIDTERM EXAM # 2: Second and higher-order ODEs Laplace transform. Definition. Main properties. HOMEWORK # 3 – L-transform. Applications of L-transform for ODES and systems of ODEs: Due at the beginning of Week XV	Improper integrals with infinite limits.	Definition and main properties of the Laplace transform.
Week XIII	Ahmad and Ambrosetti 2014, Ch. 11	Theorem(s) for inverse L- transforms	Derivatives of functions of complex variables.	Apply the theorem(s) for inverse L-transform.
Week XIV	Ahmad and Ambrosetti 2014, Ch. 11	Applications of L-transform to ODEs. Applications of L-transform to systems of ODEs.	Properties of the L- transform and inverse L-transform.	Apply the Laplace transform as solution technique.
Week XV	Ahmad and Ambrosetti 2014	Applications of L-transform to ODEs and systems of ODEs. Overview of the solutions methods discussed.	Solutions methods discussed.	Apply the L-transform. Apply all solutions methods discussed.
Week XVI	Ahmad and Ambrosetti 2014	Collect HOMEWORK # 3 Overview of the solutions methods discussed.	Solutions methods discussed.	Apply all solutions methods discussed.