Difference between revisions of "MAT3613"

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==Course Catalog==
 +
[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory 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.
 +
 +
==Text==
 +
[https://login.libweb.lib.utsa.edu/login?url=http://link.springer.com/content/pdf/10.1007%2F978-3-319-02129-4.pdf Ahmad, S., & Ambrosetti, A. (2014). textbook on Ordinary Differential Equations (Vol. 73). Springer]
 
==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
−
* [[OrderOfDE|Order of Differential Equations]]
+
* [[Order of Differential Equations]]
 
||
 
||
−
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Explain the basic notion of the order of a differential equation.
 
* Explain the basic notion of the order of a differential equation.
Line 17: Line 28:
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
−
* [[SolutionsOfDE|Solutions of Differential Equations]]
+
* [[Solutions of Differential Equations]]
 
||
 
||
−
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Explain the basic notion of solutions of differential equations.
 
* Explain the basic notion of solutions of differential equations.
Line 27: Line 42:
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
−
* [[InitialValueProblem|Initial Value Problems (IVP)]]
+
* [[Initial Value Problem|Initial Value Problem (IVP)]]
 
||
 
||
−
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Explain the basic notion of the initial values problem.
 
* Explain the basic notion of the initial values problem.
Line 37: Line 56:
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
−
* [[CauchyProblem|Cauchy Problem]]
+
* [[Cauchy Problem]]
 
||
 
||
−
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Explain the Cauchy Problem
 
* Explain the Cauchy Problem
−
* Explain the basic notion of existence and uniqueness of the Cauchy Problem.
+
* Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
 
|-
 
|-
 
|Week I
 
|Week I
Line 48: Line 71:
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
−
* [[SeparationOfVariables|Separation of Variables]]
+
* [[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|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|Linear Differential Equations (1st Order)]]
 
||
 
||
−
* Integration techniques.
+
* 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.
+
* 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
 
|Week II
Line 58: Line 117:
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
||
 
||
−
* Homogeneous differential equations.
+
* [[Integrating Factor]]
−
* Linear differential equations of first order; integrating factors.
+
||
 +
* [[Linear Differential Equations|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)]]
 
||
 
||
−
* Integration techniques.
+
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
 
||
 
||
−
* 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.
+
* Determine Bernoulli of the first order.  
 +
* Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
 
|-
 
|-
 
|Week III
 
|Week III
Line 69: Line 139:
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
||
 
||
−
* Bernoulli equations.
+
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
−
* Exact differential equations. The integrating factor for exact equations.
 
 
||
 
||
−
* Integration techniques. Partial derivatives. Linear first- order differential equations.
+
* [[Integrating Factor]] for exact equations.
 +
* [[Partial Derivatives]]
 
||
 
||
−
* 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.
+
* 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
 
|Week IV
Line 80: Line 152:
 
* Ahmad and Ambrosetti 2014, Chaps. 1-3
 
* 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).
 
* Overview of the solutions methods discussed so far (Chapters 1-3).
 
||
 
||
−
* Integration techniques. Partial derivatives. Integrating factor methods.
+
* Integration techniques
−
* First-order differential equations.
+
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 +
* [[Partial Derivatives]]
 +
* First-order differential equations:
 +
:- [[Separation of Variables (1st Order)]]
 +
:- [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
 +
:- [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
 +
:- [[Bernoulli Equations (1st Order)]]
 +
:- [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
 
||
 
||
 
* Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact.
 
* Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact.
Line 93: Line 174:
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
||
 
||
−
* MDTERM EXAM # 1:
+
* [[Linear Independence of Functions]].
−
* First-order ODEs Linear independence and Wronskian.
 
 
||
 
||
−
* Linear dependence, independenc e of vectors.
+
* [[Linear Independence of Vectors]].
−
* Determinant s.
 
 
||
 
||
−
* Linear dependence and independence of functions. Wronskian of two functions. Wronskian of two solutions of linear second-order ODEs.
+
* Understanding of Linear Independence of Functions.
 +
|-
 +
|Week V
 +
||
 +
* Ahmad and Ambrosetti 2014, Ch. 5
 +
||
 +
* [[Linear Independence of Functions|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 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
 
|Week VI
Line 105: Line 207:
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
||
 
||
−
* Reduction of the order. Linear homogeneous differential equations. Abel’s theorem.
+
* [[Reduction of the Order]]
−
* 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.
+
* [[Wronskian]].
−
* Algebraic equations.
+
* [[Quadratic Equations]].
−
* Determinant s.
+
* [[Linear Differential 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
 +
||
 +
* [[Homogeneous Differential Equations|Linear Homogeneous Equations]]
 +
||
 +
* [[Wronskian]].
 +
* [[Quadratic Equations]].
 +
* [[Linear Differential 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|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|Linear Differential Equations (1st Order)]].
 +
* [[Solutions of Linear Systems]].
 
||
 
||
−
* 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.
 +
|-
 +
|Week VI
 +
||
 +
* Ahmad and Ambrosetti 2014, Ch. 5
 +
||
 +
* [[Linear Differential Equations|Linear Non-homogeneous Equations]]
 +
||
 +
* [[Wronskian]].
 +
* [[Quadratic Equations]].
 +
* [[Linear Differential 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)|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.
 
* Apply of the variation of parameters technique for second-order ODEs.
 
|-
 
|-
Line 121: Line 290:
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
||
 
||
−
* Variation of parameters (continued)
+
* [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]] (continued)
−
* Method of undetermined coefficients
+
||
 +
* 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)]]
 
||
 
||
−
* Variation of parameters. Method of undetermined coefficients.
+
* [[Quadratic Equations]].
 +
* [[Systems of Linear Equations]].
 
||
 
||
−
* Apply variation of parameters and method of undetermined coefficients techniques for second-order ODEs.
+
* Apply method of undetermined coefficients technique for second-order ODEs.
 
|-
 
|-
 
|Week VIII
 
|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|Variation of Parameters (Higher Order)]]
 +
||
 +
* [[Variation Of Parameters|Variation of Parameters (2nd Order)]]
 +
||
 +
* Apply variation of parameters technique for higher-order ODEs
 +
|-
 +
|Week VIII
 +
||
 +
* Ahmad and Ambrosetti 2014, Ch. 5
 
||
 
||
−
* SPRING BREAK
+
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
 
||
 
||
−
 
+
* [[Method of Undetermined Coefficients (2nd Order)]]
 
||
 
||
−
 
+
* Apply method of undetermined coefficients technique for higher-order ODEs
 
|-
 
|-
 
|Week IX
 
|Week IX
 
||
 
||
−
 
+
* Ahmad and  Ambrosetti 2014, Ch. 6
 +
||
 +
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
 +
||
 +
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
 +
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]].
 +
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]].
 +
||
 +
* Methods for linear higher-order ODEs
 +
|-
 +
|Week X
 
||
 
||
−
* Preparation for remote instruction.
+
* 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|Variation of Parameters (Higher Order)]]
 +
:- [[Method of Undetermined Coefficients|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
 
|Week X
 
||
 
||
−
* Ahmad and Ambrosetti 2014, Ch. 5
+
* Ahmad and Ambrosetti 2014, Chaps. 10
 
||
 
||
−
* Higher order ODEs.
+
* [[Power Series Solutions]]
 
||
 
||
−
* Methods for higher-order ODEs.
+
* [[Power Series Induction]]
−
* Variation of parameters. Method of undetermined coefficients.
 
 
||
 
||
−
* Apply variation of parameters and method of undetermined coefficients techniques for higher-order ODEs
+
Apply power series method to evaluate solutions of first-order and second-order ODEs.
 
|-
 
|-
 
|Week XI
 
|Week XI
 
||
 
||
−
* Ahmad and Ambrosetti 2014, Chaps. 5, 6, 10
+
* Ahmad and Ambrosetti 2014, Chaps. 10
 
||
 
||
−
* Overview of the solutions methods for second and higher order differential equations.
+
* [[Power Series Solutions]] (continued)
−
* Collect HOMEWORK # 2 (extended deadline)
 
 
||
 
||
−
* Direct methods for second and higher-order ODEs.
+
* [[Power Series Induction]]
 
||
 
||
−
* Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
+
Apply power series method to evaluate solutions of first-order and second-order ODEs.
 
|-
 
|-
 
|Week XII
 
|Week XII
Line 174: Line 399:
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
||
 
||
−
* MIDTERM EXAM # 2:
+
* [[Laplace Transform]]
−
* 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.
+
* [[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 Laplace transform.
+
* Definition and main properties of the L-transform.
 
|-
 
|-
 
|Week XIII
 
|Week XIII
Line 188: Line 412:
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
||
 
||
−
* Theorem(s) for inverse L- transforms
+
* [[Inverse Laplace Transform]]
 
||
 
||
−
* Derivatives of functions of complex variables.
+
* [[Laplace Transform]]
 +
* [[Complex Derivatives]]
 
||
 
||
 
* Apply the theorem(s) for inverse L-transform.
 
* Apply the theorem(s) for inverse L-transform.
Line 198: Line 423:
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
* Ahmad and Ambrosetti 2014, Ch. 11
 
||
 
||
−
* Applications of L-transform to ODEs.
+
* [[Laplace Transform to ODEs]]
−
* Applications of L-transform to systems of ODEs.
 
 
||
 
||
−
* Properties of the L- transform and inverse L-transform.
+
* [[Linear Differential Equations|Linear Equations]]
 +
* [[Laplace Transform]]
 +
* [[Inverse Laplace Transform]]
 
||
 
||
 
* Apply the Laplace transform as solution technique.
 
* Apply the Laplace transform as solution technique.
 
|-
 
|-
−
|Week XV
+
|Week XIV
 
||
 
||
−
* Ahmad and Ambrosetti 2014
+
* Ahmad and Ambrosetti 2014, Ch. 11
 
||
 
||
−
* Applications of L-transform to ODEs and systems of ODEs.
+
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
−
* Overview of the solutions methods discussed.
 
 
||
 
||
−
* Solutions methods discussed.
+
* [[Solutions of Linear Systems]].
 +
* [[Laplace Transform]].
 +
* [[Inverse Laplace Transform]].
 
||
 
||
−
* Apply the L-transform. Apply all solutions methods discussed.
+
* Apply the Laplace transform as solution technique.
 
|-
 
|-
−
|Week XVI
+
|Week XV
 
||
 
||
 
* Ahmad and Ambrosetti 2014
 
* Ahmad and Ambrosetti 2014
 
||
 
||
−
* Collect HOMEWORK # 3 Overview of the solutions methods discussed.
+
* Overview of the solutions methods discussed.
 
||
 
||
−
* Solutions methods discussed.
+
* [[Separation of Variables (1st Order)]]
 +
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
 +
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
 +
* [[Integrating Factor]]
 +
* [[Bernoulli Equations (1st Order)]]
 +
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
 +
* [[Reduction of the Order]]
 +
* [[Method of Undetermined Coefficients (2nd Order)]]
 +
* [[Non-linear 2nd Order ODEs]]
 +
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
 +
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
 +
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
 +
* [[Power Series Solutions]]
 +
* [[Laplace Transform to ODEs]]
 +
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
 
||
 
||
 
* Apply all solutions methods discussed.
 
* Apply all solutions methods discussed.
 
|}
 
|}

Latest revision as of 12:24, 27 November 2021

Course Catalog

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.

Text

Ahmad, S., & Ambrosetti, A. (2014). textbook on Ordinary Differential Equations (Vol. 73). Springer

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week I
  • Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
  • Integration techniques
- Direct Integration
- Integration by Substitution
- 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
- 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
  • Understanding of Linear Independence of Functions.
Week V
  • Ahmad and Ambrosetti 2014, Ch. 5
  • Understanding of Linear Dependence of Functions.
Week V
  • Ahmad and Ambrosetti 2014, Ch. 5
  • 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
  • Apply of the reduction of the order technique for second-order ODEs with a given solution.
Week VI
  • Ahmad and Ambrosetti 2014, Ch. 5
  • 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
  • Determine Wronskian for a second-order ODE with 2 given solutions.
Week VI
  • Ahmad and Ambrosetti 2014, Ch. 5
  • Determine fundamental solutions.
Week VI
  • Ahmad and Ambrosetti 2014, Ch. 5
  • 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
  • Integration techniques
- Direct Integration
- Integration by Substitution
- Integration by Parts
- Partial Fractions
  • Apply of the variation of parameters technique for second-order ODEs.
Week VII
  • Ahmad and Ambrosetti 2014, Ch. 5
  • Integration techniques
- Direct Integration
- Integration by Substitution
- Integration by Parts
- Partial Fractions
  • Apply variation of parameters technique for second-order ODEs.
Week VII
  • Ahmad and Ambrosetti 2014, Ch. 5
  • Apply method of undetermined coefficients technique for second-order ODEs.
Week VIII
  • Ahmad and Ambrosetti 2014, Ch. 5
- 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
  • Apply variation of parameters technique for higher-order ODEs
Week VIII
  • Ahmad and Ambrosetti 2014, Ch. 5
  • Apply method of undetermined coefficients technique for higher-order ODEs
Week IX
  • Ahmad and Ambrosetti 2014, Ch. 6
  • 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.
- 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

Apply power series method to evaluate solutions of first-order and second-order ODEs.

Week XI
  • Ahmad and Ambrosetti 2014, Chaps. 10

Apply power series method to evaluate solutions of first-order and second-order ODEs.

Week XII
  • Ahmad and Ambrosetti 2014, Ch. 11
  • Definition and main properties of the L-transform.
Week XIII
  • Ahmad and Ambrosetti 2014, Ch. 11
  • Apply the theorem(s) for inverse L-transform.
Week XIV
  • Ahmad and Ambrosetti 2014, Ch. 11
  • Apply the Laplace transform as solution technique.
Week XIV
  • Ahmad and Ambrosetti 2014, Ch. 11
  • Apply the Laplace transform as solution technique.
Week XV
  • Ahmad and Ambrosetti 2014
  • Overview of the solutions methods discussed.
  • Apply all solutions methods discussed.
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