MAT3613

MAT 3613. Differential Equations I. (3-0) 3 Credit Hours.

Prerequisite: Completion of or concurrent enrollment in MAT2233. Basic notions of differential equations, solution of first-order equations and linear equations with constant coefficients, nth-order initial value problems, Laplace transforms, and may include additional topics such as power series solutions of differential equations, linear systems, and stability. Generally offered: Fall, Spring, Summer. Differential Tuition: $150.

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Order of Differential Equations	Integration techniques - Direct Integration - Integration by Substitutions - Integration by Parts - Partial Fractions	Explain the basic notion of the order of a differential equation.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Solutions of Differential Equations	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Explain the basic notion of solutions of differential equations.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Initial Value Problem (IVP)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Explain the basic notion of the initial values problem.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Cauchy Problem	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Explain the Cauchy Problem Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Separation of Variables (1st Order)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	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 (1st Order)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Determine homogeneous differential equations of the first order. Apply direct methods to evaluate exact solutions of homogeneous differential equations of the first order (substitutions). Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
Week II	Ahmad and Ambrosetti 2014, Chaps. 1 and 3	Linear Differential Equations (1st Order)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Determine linear differential equations of the first order. Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
Week II	Ahmad and Ambrosetti 2014, Chaps. 1 and 3	Integrating Factor	Linear Differential Equations (1st Order)	Apply integrating factor to solve linear differential equations of the first order. 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 (1st Order)	Linear Differential Equations (1st Order)	Determine Bernoulli of the first order. Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
Week III	Ahmad and Ambrosetti 2014, Ch. 3	Exact Differential Equations (1st Order)	Integrating Factor for exact equations. Partial Derivatives	Determine Exact Differential Equations of the first order. Apply direct methods to evaluate exact solutions of Exact Differential Equations of the first order. Use the integrating factor technique for exact equations.
Week IV	Ahmad and Ambrosetti 2014, Chaps. 1-3	Overview of the solutions methods discussed so far (Chapters 1-3).	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions Partial Derivatives First-order differential equations: - Separation of Variables (1st Order) - Homogeneous Differential Equations (1st Order) - Linear Differential Equations (1st Order) - Bernoulli Equations (1st Order) - Exact Differential Equations (1st Order)	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	Linear Independence of Functions.	Linear Independence of Vectors.	Understanding of Linear Independence of Functions.
Week V	Ahmad and Ambrosetti 2014, Ch. 5	Linear Dependence of Functions.	Linear Dependence of Vectors.	Understanding of Linear Dependence of Functions.
Week V	Ahmad and Ambrosetti 2014, Ch. 5	Wronskian	Linear Independence of Functions. Linear Dependence of Functions. Determinant.	Showing linear independence of two functions using the Wronskian. Showing linear independence of two solutions of Linear Second-Order ODEs using the Wronskian.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Reduction of the Order	Wronskian. Quadratic Equations. Linear Differential Equations (1st Order). Solutions of Linear Systems.	Apply of the reduction of the order technique for second-order ODEs with a given solution.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Linear Homogeneous Equations	Wronskian. Quadratic Equations. Linear Differential Equations (1st Order). Solutions of Linear Systems.	Determine homogeneous classes of differential equations of the second and higher order. Determine linear and non-linear classes of differential equations of the second and higher order.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Abel’s Theorem	Wronskian. Quadratic Equations. Linear Differential Equations (1st Order). Solutions of Linear Systems.	Determine Wronskian for a second-order ODE with 2 given solutions.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Fundamental Solutions	Wronskian. Quadratic Equations. Linear Differential Equations (1st Order). Solutions of Linear Systems.	Determine fundamental solutions.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Linear Non-homogeneous Equations	Wronskian. Quadratic Equations. Linear Differential Equations (1st Order). Solutions of Linear Systems.	Determine non-homogeneous classes of differential equations of the second and higher order. Determine linear and non-linear classes of differential equations of the second and higher order
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Variation of Parameters (2nd Order)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions Quadratic Equations. Solutions of Linear Systems.	Apply of the variation of parameters technique for second-order ODEs.
Week VII	Ahmad and Ambrosetti 2014, Ch. 5	Variation of Parameters (2nd Order) (continued)	Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions Quadratic Equations. Solutions of Linear Systems.	Apply variation of parameters technique for second-order ODEs.
Week VII	Ahmad and Ambrosetti 2014, Ch. 5	Method of Undetermined Coefficients (2nd Order)	Quadratic Equations. Systems of Linear Equations.	Apply method of undetermined coefficients technique for second-order ODEs.
Week VIII	Ahmad and Ambrosetti 2014, Ch. 5	Non-linear 2nd Order ODEs	Algebraic Equations Reduction of the Order Integration techniques - Direct Integration - Integration by Substitution - Integration by Parts - Partial Fractions	Methods for nonlinear second-order ODEs. Apply reduction of the order method to some nonlinear second-order ODEs.
Week VIII	Ahmad and Ambrosetti 2014, Ch. 5	Variation of Parameters (Higher Order)	Variation of Parameters (2nd Order)	Apply variation of parameters technique for higher-order ODEs
Week VIII	Ahmad and Ambrosetti 2014, Ch. 5	Method of Undetermined Coefficients (Higher Order)	Method of Undetermined Coefficients (2nd Order)	Apply method of undetermined coefficients technique for higher-order ODEs
Week IX	Ahmad and Ambrosetti 2014, Ch. 6	Linear Differential Equations (Higher Order)	Linear Differential Equations (1st Order) Variation of Parameters (Higher Order). Method of Undetermined Coefficients (Higher Order).	Methods for linear higher-order ODEs
Week X	Ahmad and Ambrosetti 2014, Chaps. 5, 6	Overview of the solutions methods for second and higher order differential equations.	Algebraic Equations Direct methods for second and higher-order ODEs: - Variation of Parameters (Higher Order) - Method of Undetermined Coefficients (Higher Order)	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 X	Ahmad and Ambrosetti 2014, Chaps. 10	Power Series Solutions	Power Series Inducution	Apply power series method to evaluate solutions of first-order and second-order ODEs.
Week XI	Ahmad and Ambrosetti 2014, Chaps. 10	Power Series Solutions (continued)	Power Series Induction	Apply power series method to evaluate solutions of first-order and second-order ODEs.
Week XII	Ahmad and Ambrosetti 2014, Ch. 11	Laplace Transform	Functions of Single Variable. Continuity of functions of single variables. Derivatives of functions of single variables. Improper Integrals of functions of single variables with infinite limits.	Definition and main properties of the L-transform.
Week XIII	Ahmad and Ambrosetti 2014, Ch. 11	Inverse Laplace Transform	Laplace Transform Complex Derivatives	Apply the theorem(s) for inverse L-transform.
Week XIV	Ahmad and Ambrosetti 2014, Ch. 11	Laplace Transform to ODEs	Linear Equations Laplace Transform Inverse Laplace Transform	Apply the Laplace transform as solution technique.
Week XIV	Ahmad and Ambrosetti 2014, Ch. 11	Laplace Transform to Systems of ODEs	Solutions of Linear Systems. Laplace Transform. Inverse Laplace Transform.	Apply the Laplace transform as solution technique.
Week XV	Ahmad and Ambrosetti 2014	Overview of the solutions methods discussed.	Separation of Variables (1st Order) Homogeneous Differential Equations (1st Order) Linear Differential Equations (1st Order) Integrating Factor Bernoulli Equations (1st Order) Exact Differential Equations (1st Order) Reduction of the Order Method of Undetermined Coefficients (2nd Order) Non-linear 2nd Order ODEs Variation of Parameters (Higher Order) Method of Undetermined Coefficients (Higher Order) Linear Differential Equations (Higher Order) Power Series Solutions Laplace Transform to ODEs Laplace Transform to Systems of ODEs	Apply all solutions methods discussed.