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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"
Line 7: Line 14:
 
* 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 Problem (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
Line 48: Line 71:
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
 
||
 
||
* [[SeparationOfVariables|Separation of Variables (1st Order)]]
+
* [[Separation of Variables (1st Order)]]
 
||
 
||
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Determine separable differential equations of the first order.  
 
* Determine separable differential equations of the first order.  
Line 59: Line 86:
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
||
 
||
* [[HomogeneousDE|Homogeneous Differential Equations (1st Order)]]
+
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
 
||
 
||
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Determine homogeneous differential equations of the first order.  
 
* Determine homogeneous differential equations of the first order.  
Line 71: Line 102:
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 
||
 
||
* [[LinearDE|Linear Differential Equations (1st Order)]]
+
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
 
||
 
||
* Integration techniques.
+
* Integration techniques
 +
:- [[Direct Integration]]
 +
:- [[Integration by Substitution]]
 +
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 
||
 
||
 
* Determine linear differential equations of the first order.  
 
* 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 II
 +
||
 +
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
 +
||
 +
* [[Integrating Factor]]
 +
||
 +
* [[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.
 
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
 
|-
 
|-
Line 83: Line 128:
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
||
 
||
* [[BernoulliEquations|Bernoulli Equations (1st Order)]]
+
* [[Bernoulli Equations (1st Order)]]
 
||
 
||
* Integration techniques:
+
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
:- [[PartialDerivatives|Partial Derivatives]]
 
:- [[LinearDE|Linear Differential Equations (1st Order)]]
 
 
||
 
||
 
* Determine Bernoulli of the first order.  
 
* Determine Bernoulli of the first order.  
Line 96: Line 139:
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
* Ahmad and Ambrosetti 2014, Ch. 3
 
||
 
||
* [[ExactDifferentialEquations|Exact Differential Equations]]  
+
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
 
||
 
||
* The integrating factor for exact equations.
+
* [[Integrating Factor]] for exact equations.
* Integration techniques:
+
* [[Partial Derivatives]]
:- [[PartialDerivatives|Partial Derivatives]]
 
:- [[LinearDE|Linear Differential Equations (1st Order)]]
 
 
||
 
||
 
* Determine Exact Differential Equations of the first order.  
 
* Determine Exact Differential Equations of the first order.  
Line 113: Line 154:
 
* Overview of the solutions methods discussed so far (Chapters 1-3).
 
* Overview of the solutions methods discussed so far (Chapters 1-3).
 
||
 
||
* Integration techniques:
+
* Integration techniques
:- [[PartialDerivatives|Partial Derivatives]]
+
:- [[Direct Integration]]
:- [[LinearDE|Linear Differential Equations (1st Order)]]
+
:- [[Integration by Substitution]]
 
+
:- [[Integration by Parts]]
 +
:- [[Partial Fractions]]
 +
* [[Partial Derivatives]]
 
* First-order differential equations:
 
* First-order differential equations:
:- [[SeparationOfVariables|Separation of Variables (1st Order)]]
+
:- [[Separation of Variables (1st Order)]]
:- [[HomogeneousDE|Homogeneous Differential Equations (1st Order)]]
+
:- [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
:- [[LinearDE|Linear Differential Equations (1st Order)]]
+
:- [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
:- [[BernoulliEquations|Bernoulli Equations (1st Order)]]
+
:- [[Bernoulli Equations (1st Order)]]
:- [[ExactDifferentialEquations|Exact Differential Equations]]
+
:- [[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 131: Line 174:
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
* Ahmad and Ambrosetti 2014, Ch. 5
 
||
 
||
* First-order ODEs Linear independence and Wronskian.
+
* [[Linear Independence of Functions]].
 
||
 
||
* Linear dependence, independence of vectors.
+
* [[Linear Independence of Vectors]].
* Determinants.
 
 
||
 
||
* 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 142: 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 158: 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
 
 
||
 
||
* Variation of parameters. 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 and method of undetermined coefficients techniques for second-order ODEs.
+
* 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
 
|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
 +
||
 +
* Ahmad and Ambrosetti 2014, Chaps. 5, 6
 
||
 
||
* Preparation for remote instruction.
+
* 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 211: 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 225: 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 235: 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.