MAT3613

From Department of Mathematics at UTSA
Revision as of 09:35, 2 July 2020 by Johnraymond.yanez (talk | contribs) (Edited Week IV)
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Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week I
  • Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
  • Integration techniques.
  • Explain the basic notion of the order of a differential equation.
Week I
  • Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
  • Integration techniques.
  • Explain the basic notion of solutions of differential equations.
Week I
  • Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
  • Integration techniques.
  • Explain the basic notion of the initial values problem.
Week I
  • Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
  • Integration techniques.
  • 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
  • Integration techniques.
  • 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
  • Integration techniques.
  • 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
  • Integration techniques.
  • Determine linear differential equations of the first order.
  • Apply direct methods to evaluate exact solutions of 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
  • Integration techniques:
- Partial Derivatives
- 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
  • The integrating factor for exact equations.
  • Integration techniques:
- Partial Derivatives
- Linear Differential Equations (1st Order)
  • 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:
- Partial Derivatives
- Linear Differential Equations (1st Order)
  • 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
  • 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
  • First-order ODEs Linear independence and Wronskian.
  • Linear dependence, independence of vectors.
  • Determinants.
  • 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.