# MAT3613

## 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.

## 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