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==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[OrderOfDE|Order of Differential Equations]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitutions|Integration by Substitutions]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* Explain the basic notion of the order of a differential equation.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[SolutionsOfDE|Solutions of Differential Equations]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* Explain the basic notion of solutions of differential equations.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[InitialValueProblem|Initial Value Problem (IVP)]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]].
||
* Explain the basic notion of the initial values problem.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[CauchyProblem|Cauchy Problem]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* 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
||
* [[SeparationOfVariables1Ord|Separation of Variables (1st Order)]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* 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
||
* [[HomogeneousDE1Ord|Homogeneous Differential Equations (1st Order)]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* 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
||
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* 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
||
* [[IntegratingFactor|Integrating Factor]]
||
* [[LinearDE1Ord|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
||
* [[BernoulliEquations1Ord|Bernoulli Equations (1st Order)]]
||
* [[LinearDE1Ord|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
||
* [[ExactDE1Ord|Exact Differential Equations (1st Order)]]
||
* [[IntegratingFactor|Integrating Factor]] for exact equations.
* [[PartialDerivatives|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
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
* [[PartialDerivatives|Partial Derivatives]]
* First-order differential equations:
:- [[SeparationOfVariables1Ord|Separation of Variables (1st Order)]]
:- [[HomogeneousDE1Ord|Homogeneous Differential Equations (1st Order)]]
:- [[LinearDE1Ord|Linear Differential Equations (1st Order)]]
:- [[BernoulliEquations1Ord|Bernoulli Equations (1st Order)]]
:- [[ExactDE1Ord|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
||
* [[LinearIndependenceOfFunctions|Linear Independence of Functions]].
||
* [[LinearIndependenceOfVectors|Linear Independence of Vectors]].
||
* Understanding of Linear Independence of Functions. 
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[LinearDependenceOfFunctions|Linear Dependence of Functions]].
||
* [[LinearDependenceOfVectors|Linear Dependence of Vectors]].
||
* Understanding of Linear Dependence of Functions. 
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Wronskian|Wronskian]]
||
* [[LinearIndependenceOfFunctions|Linear Independence of Functions]].
* [[LinearDependenceOfFunctions|Linear Dependence of Functions]].
* [[Determinant|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
||
* [[ReductionOfTheOrder|Reduction of the Order]]
||
* [[Wronskian|Wronskian]].
* [[QuadraticEquations|Quadratic Equations]].
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Apply of the reduction of the order technique for second-order ODEs with a given solution.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[LinearHomogeneousEquations|Linear Homogeneous Equations]]
||
* [[Wronskian|Wronskian]].
* [[QuadraticEquations|Quadratic Equations]].
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* 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
||
* [[AbelsTheorem|Abel’s Theorem]]
||
* [[Wronskian|Wronskian]].
* [[QuadraticEquations|Quadratic Equations]].
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Determine Wronskian for a second-order ODE with 2 given solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[FundamentalSolutions|Fundamental Solutions]]
||
* [[Wronskian|Wronskian]].
* [[QuadraticEquations|Quadratic Equations]].
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Determine fundamental solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[LinearNonhomogeneousEquations|Linear Non-homogeneous Equations]]
||
* [[Wronskian|Wronskian]].
* [[QuadraticEquations|Quadratic Equations]].
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* 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
||
* [[VariationOfParameters2Ord|Variation of Parameters (2nd Order)]]
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
* [[QuadraticEquations|Quadratic Equations]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Apply of the variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[VariationOfParameters2Ord|Variation of Parameters (2nd Order)]] (continued)
||
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
* [[QuadraticEquations|Quadratic Equations]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Apply variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[MethodOfUndeterminedCoefficients2Ord|Method of Undetermined Coefficients (2nd Order)]]
||
* [[QuadraticEquations|Quadratic Equations]].
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
||
* Apply method of undetermined coefficients technique for second-order ODEs.
|-
|Week VIII
||
* Ahmad and  Ambrosetti 2014, Ch. 5
||
* [[Nonlinear2OrdODEs|Non-linear 2nd Order ODEs]]
||
* [[AlgebraicEquations|Algebraic Equations]]
* [[ReductionOfTheOrder|Reduction of the Order]]
* Integration techniques
:- [[DirectIntegration|Direct Integration]]
:- [[IntegrationBySubstitution|Integration by Substitution]]
:- [[IntegrationByParts|Integration by Parts]]
:- [[IntegrationOfRationalFunctions|Integration of Rational Functions]]
||
* 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
||
* [[VariationOfParametershOrd|Variation of Parameters (Higher Order)]]
||
* [[VariationOfParameters2Ord|Variation of Parameters (2nd Order)]]
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* Apply variation of parameters technique for higher-order ODEs
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[MethodOfUndeterminedCoefficientshOrd|Method of Undetermined Coefficients (Higher Order)]]
||
* Methods for higher-order ODEs
* [[MethodOfUndeterminedCoefficients2Ord|Method of Undetermined Coefficients (2nd Order)]]
||
* Apply method of undetermined coefficients technique for higher-order ODEs
|-
|Week IX
||
* Ahmad and  Ambrosetti 2014, Ch. 6
||
* [[LinearDEhOrd|Linear Differential Equations (Higher Order)]]
||
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]]
* [[VariationOfParametershOrd|Variation of Parameters (Higher Order)]].
* [[MethodOfUndeterminedCoefficientshOrd|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.
||
* [[AlgebraicEquations|Algebraic Equations]]
* Direct methods for second and higher-order ODEs:
:- [[VariationOfParametershOrd|Variation of Parameters (Higher Order)]]
:- [[MethodOfUndeterminedCoefficientshOrd|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
||
* [[PowerSeriesSolutions|Power Series Solutions]]
||
* [[PowerSeriesInducution|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
||
* [[PowerSeriesSolutions|Power Series Solutions]] (continued)
||
* [[PowerSeriesInducution|Power Series Inducution]]
||
Apply power series method to evaluate solutions of first-order and second-order ODEs.
|-
|Week XII
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[LTransform|Laplace Transform]]
||
* [[Functions|Functions]] of Single Variable.
* [[Continuity|Continuity]] of functions of single variables.
* [[Derivatives|Derivatives]] of functions of single variables.
* [[ImproperIntegrals|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
||
* [[InverseLTransform|Inverse Laplace Transform]]
||
* [[LTransform|Laplace Transform]]
* [[ComplexDerivatives|ComplexDerivatives]]
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* Apply the theorem(s) for inverse L-transform.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[LTransformToODEs|L-Transform to ODEs]]
||
* [[LinearEquations|Linear Equations]]
* [[LTransform|Laplace Transform]]
* [[InverseLTransform|Inverse Laplace Transform]]
||
* Apply the Laplace transform as solution technique.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[LTransformToSystemsOfODEs|L-Transform to Systems of ODEs]]
||
* [[SystemsOfLinearEquations|Systems of Linear Equations]].
* [[LTransform|Laplace Transform]].
* [[InverseLTransform|Inverse Laplace Transform]].
||
* Apply the Laplace transform as solution technique.
|-
|Week XV
||
* Ahmad and Ambrosetti 2014
||
* Overview of the solutions methods discussed.
||
* [[SeparationOfVariables1Ord|Separation of Variables (1st Order)]]
* [[HomogeneousDE1Ord|Homogeneous Differential Equations (1st Order)]]
* [[LinearDE1Ord|Linear Differential Equations (1st Order)]]
* [[IntegratingFactor|Integrating Factor]]
* [[BernoulliEquations1Ord|Bernoulli Equations (1st Order)]]
* [[ExactDE1Ord|Exact Differential Equations (1st Order)]]
* [[ReductionOfTheOrder|Reduction of the Order]]
* [[MethodOfUndeterminedCoefficients2Ord|Method of Undetermined Coefficients (2nd Order)]]
* [[Nonlinear2OrdODEs|Non-linear 2nd Order ODEs]]
* [[VariationOfParametershOrd|Variation of Parameters (Higher Order)]]
* [[MethodOfUndeterminedCoefficientshOrd|Method of Undetermined Coefficients (Higher Order)]]
* [[LinearDEhOrd|Linear Differential Equations (Higher Order)]]
* [[PowerSeriesSolutions|Power Series Solutions]]
* [[LTransformToODEs|L-Transform to ODEs]]
* [[LTransformToSystemsOfODEs|L-Transform to Systems of ODEs]]
||
* Apply all solutions methods discussed.
|}