Difference between revisions of "MAT3613"

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|Week I
 
|Week I
 
||
 
||
* Ahmad and Ambrosetti 2014,
+
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
* Chaps. 1,
+
 
* 2, 3
 
* nan
 
* nan
 
* nan
 
 
||
 
||
 
* Differential equations: what they are and why they are important; order of a differential equation; solutions; initial value problems.
 
* 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.
 
* Theory of first order differential equations; Cauchy problem; existence and uniqueness.
 
* The method of separation of variables; examples of separable differential equations.
 
* The method of separation of variables; examples of separable differential equations.
* nan
+
* HOMEWORK # 1 – First Order ODEs: Due at the beginning of Week IV
* HOMEWORK # 1 – First
 
* Order ODEs: Due at the beginning of Week IV
 
 
||
 
||
 
* Integration techniques.
 
* Integration techniques.
* nan
 
* nan
 
* nan
 
* nan
 
* nan
 
 
||
 
||
 
* Explain the basic notions related to differential equations: the order of a differential equation, solutions, basic concepts of initial values, existence and uniqueness.
 
* 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.
 
* Determine separable differential equations of the first order. Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
* nan
 
* nan
 
* nan
 
* nan
 
 
|-
 
|-
 
|Week II
 
|Week II
 
||
 
||
* Ahmad and Ambrosetti 2014,
+
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
* Chaps. 1
 
* and 3
 
 
||
 
||
 
* Homogeneous differential equations.
 
* Homogeneous differential equations.
 
* Linear differential equations of first order; integrating factors.
 
* Linear differential equations of first order; integrating factors.
* nan
 
 
||
 
||
 
* Integration techniques.
 
* Integration techniques.
* nan
 
* nan
 
 
||
 
||
* 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
+
* 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.
* problems.
 
* nan
 
 
|-
 
|-
 
|Week III
 
|Week III
 
||
 
||
* Ahmad and Ambrosetti 2014, Ch.
+
* Ahmad and Ambrosetti 2014, Ch. 3
* 3
 
* nan
 
 
||
 
||
 
* Bernoulli equations.
 
* Bernoulli equations.
 
* Exact differential equations. The integrating factor for exact equations.
 
* Exact differential equations. The integrating factor for exact equations.
* nan
 
 
||
 
||
 
* Integration techniques. Partial derivatives. Linear first- order differential equations.
 
* Integration techniques. Partial derivatives. Linear first- order differential equations.
* nan
 
* nan
 
 
||
 
||
* 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
+
* 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.
* exact equations.
 
* nan
 
 
|-
 
|-
 
|Week IV
 
|Week IV
 
||
 
||
* Ahmad and Ambrosetti 2014,
+
* Ahmad and Ambrosetti 2014, Chaps. 1-3
* Chaps. 1-3
 
 
||
 
||
 
* Collect HOMEWORK # 1 Firs-order ODEs not solved for the first derivative: Clairaut equations, Lagrange equations.
 
* Collect HOMEWORK # 1 Firs-order ODEs not solved for the first derivative: Clairaut equations, Lagrange equations.
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||
 
||
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
* nan
 
 
||
 
||
 
* MDTERM EXAM # 1:
 
* MDTERM EXAM # 1:
Line 95: Line 64:
 
||
 
||
 
* Linear dependence and independence of functions. Wronskian of two functions. Wronskian of two solutions of linear second-order ODEs.
 
* Linear dependence and independence of functions. Wronskian of two functions. Wronskian of two solutions of linear second-order ODEs.
* nan
 
 
|-
 
|-
 
|Week VI
 
|Week VI
 
||
 
||
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
* nan
 
* nan
 
* nan
 
* nan
 
* nan
 
 
||
 
||
 
* Reduction of the order. Linear homogeneous differential equations. Abel’s theorem.
 
* Reduction of the order. Linear homogeneous differential equations. Abel’s theorem.
 
* Fundamental solutions. Linear nonhomogeneous equations; variation of parameters.
 
* Fundamental solutions. Linear nonhomogeneous equations; variation of parameters.
* nan
+
* HOMEWORK # 2 – Second and higher order ODEs: Due at the beginning of Week X (extended later)
* HOMEWORK # 2 –
 
* Second and higher order ODEs: Due at the beginning of Week X
 
* (extended later)
 
 
||
 
||
* Wronskian. Algebraic equations. Determinant s.
+
* Wronskian.
* nan
+
* Algebraic equations.
* nan
+
* Determinant s.
* nan
 
* nan
 
* nan
 
 
||
 
||
 
* 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 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.
 
* Determine fundamental solutions.
 
* Apply of the variation of parameters technique for second-order ODEs.
 
* Apply of the variation of parameters technique for second-order ODEs.
* nan
 
* nan
 
* nan
 
 
|-
 
|-
 
|Week VII
 
|Week VII
 
||
 
||
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
* nan
 
 
||
 
||
 
* Variation of parameters (continued)
 
* Variation of parameters (continued)
 
* Method of undetermined coefficients
 
* Method of undetermined coefficients
 
||
 
||
* Variation of parameters. Method of undetermine d coefficients.
+
* Variation of parameters. Method of undetermined coefficients.
* nan
 
 
||
 
||
 
* Apply variation of parameters and method of undetermined coefficients techniques for second-order ODEs.
 
* Apply variation of parameters and method of undetermined coefficients techniques for second-order ODEs.
* nan
 
 
|-
 
|-
 
|Week VIII
 
|Week VIII
 
||
 
||
* asdfa
+
 
 
||
 
||
* SPRING BRAKE
+
* SPRING BREAK
 
||
 
||
* asdfas
+
 
 
||
 
||
* asdfsa
+
 
 
|-
 
|-
 
|Week IX
 
|Week IX
 
||
 
||
* adfaf
+
 
 
||
 
||
 
* Preparation for remote instruction.
 
* Preparation for remote instruction.
 
||
 
||
* asdfas
+
 
 
||
 
||
* asdfas
+
 
 
|-
 
|-
 
|Week X
 
|Week X
 
||
 
||
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
* nan
 
* nan
 
 
||
 
||
 
* Higher order ODEs.
 
* Higher order ODEs.
* nan
 
* nan
 
 
||
 
||
 
* Methods for higher-order ODEs.
 
* Methods for higher-order ODEs.
* Variation of parameters. Method of undetermine d coefficients.
+
* Variation of parameters. Method of undetermined coefficients.
* nan
 
 
||
 
||
 
* Apply variation of parameters and method of undetermined coefficients techniques for higher-order ODEs
 
* Apply variation of parameters and method of undetermined coefficients techniques for higher-order ODEs
* nan
 
* nan
 
 
|-
 
|-
 
|Week XI
 
|Week XI
 
||
 
||
* Ahmad and Ambrosetti 2014,
+
* Ahmad and Ambrosetti 2014, Chaps. 5, 6, 10
* Chaps. 5,
 
* 6, 10
 
 
||
 
||
 
* Overview of the solutions methods for second and higher order differential equations.
 
* Overview of the solutions methods for second and higher order differential equations.
 
* Collect HOMEWORK # 2 (extended deadline)
 
* Collect HOMEWORK # 2 (extended deadline)
* nan
 
 
||
 
||
 
* Direct methods for second and higher-order ODEs.
 
* Direct methods for second and higher-order ODEs.
* nan
 
* nan
 
 
||
 
||
 
* Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
 
* Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
* nan
 
* nan
 
 
|-
 
|-
 
|Week XII
 
|Week XII
 
||
 
||
* Ahmad and Ambrosetti 2014, Ch.
+
* Ahmad and Ambrosetti 2014, Ch. 11
* 11
 
* nan
 
* nan
 
* nan
 
* nan
 
 
||
 
||
* MDTERM EXAM # 2:
+
* MIDTERM EXAM # 2:
 
* Second and higher-order ODEs
 
* Second and higher-order ODEs
 
* Laplace transform. Definition.
 
* Laplace transform. Definition.
 
* Main properties.
 
* Main properties.
* HOMEWORK # 3 – L-
+
* HOMEWORK # 3 – L-transform. Applications of L-transform for ODES and systems of ODEs: Due at the beginning of Week XV
* transform. Applications of L-transform for ODES and systems of ODEs: Due at the beginning of Week XV
 
 
||
 
||
 
* Improper integrals with infinite limits.
 
* Improper integrals with infinite limits.
* nan
 
* nan
 
* nan
 
* nan
 
* nan
 
 
||
 
||
 
* Definition and main properties of the Laplace transform.
 
* Definition and main properties of the Laplace transform.
* nan
 
* nan
 
* nan
 
* nan
 
* nan
 
 
|-
 
|-
 
|Week XIII
 
|Week XIII
 
||
 
||
* Ahmad and Ambrosetti 2014, Ch.
+
* Ahmad and Ambrosetti 2014, Ch. 11
* 11
 
 
||
 
||
 
* Theorem(s) for inverse L- transforms
 
* Theorem(s) for inverse L- transforms
* nan
 
 
||
 
||
 
* Derivatives of functions of complex variables.
 
* Derivatives of functions of complex variables.
* nan
 
 
||
 
||
 
* Apply the theorem(s) for inverse L-transform.
 
* Apply the theorem(s) for inverse L-transform.
* nan
 
 
|-
 
|-
 
|Week XIV
 
|Week XIV
 
||
 
||
* Ahmad and Ambrosetti 2014, Ch.
+
* Ahmad and Ambrosetti 2014, Ch. 11
* 11
 
 
||
 
||
 
* Applications of L-transform to ODEs.
 
* Applications of L-transform to ODEs.
Line 250: Line 166:
 
||
 
||
 
* Properties of the L- transform and inverse L-transform.
 
* Properties of the L- transform and inverse L-transform.
* nan
 
 
||
 
||
 
* Apply the Laplace transform as solution technique.
 
* Apply the Laplace transform as solution technique.
* nan
 
 
|-
 
|-
 
|Week XV
 
|Week XV
 
||
 
||
 
* Ahmad and Ambrosetti 2014
 
* Ahmad and Ambrosetti 2014
* nan
 
 
||
 
||
 
* Applications of L-transform to ODEs and systems of ODEs.
 
* Applications of L-transform to ODEs and systems of ODEs.
Line 264: Line 177:
 
||
 
||
 
* Solutions methods discussed.
 
* Solutions methods discussed.
* nan
 
 
||
 
||
 
* Apply the L-transform. Apply all solutions methods discussed.
 
* Apply the L-transform. Apply all solutions methods discussed.
* nan
 
 
|-
 
|-
 
|Week XVI
 
|Week XVI
 
||
 
||
 
* Ahmad and Ambrosetti 2014
 
* Ahmad and Ambrosetti 2014
* nan
 
 
||
 
||
 
* Collect HOMEWORK # 3 Overview of the solutions methods discussed.
 
* Collect HOMEWORK # 3 Overview of the solutions methods discussed.
* nan
 
 
||
 
||
 
* Solutions methods discussed.
 
* Solutions methods discussed.
* nan
 
 
||
 
||
 
* Apply all solutions methods discussed.
 
* Apply all solutions methods discussed.
* nan
 
 
|}
 
|}

Revision as of 09:42, 30 June 2020

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.