MATH 2770 - Differential Equations

Credit Hours: 4.00

Objectives:

1. Identify separable ODEs and employ appropriate techniques to solve separable ODEs.
2. Identify exact ODEs and employ appropriate techniques to solve exact ODEs.
3. Identify first-order linear ODEs and employ appropriate techniques to solve first-order linear ODEs.
4. Identify Bernoulli ODEs and employ appropriate techniques to solve Bernoulli ODEs.
5. Use numerical methods, including the improved Euler method or RK4 to calculate numerical solutions over intervals to any of the above types of ODEs with initial conditions.

Outcome 2: Upon completion of this course, students will be able to apply differential equations to solve mathematical problems.

Objectives:

1. Solve continuous growth and decay models.
2. Solve cooling and warming models.
3. Solve mixture models.
4. Solve series circuit models.

Outcome 3: Upon completion of this course, students will be able to solve homogeneous and non-homogeneous higher order linear equations using the methods of undetermined coefficients and variation of parameters.

Objectives

1. Solve homogeneous higher order ODEs using the superposition principle.
2. Solve homogeneous higher order ODEs using the operator notation and by the annihilator approach.
3. Solve non-homogeneous higher order ODEs using the method of undetermined coefficients.
4. Solve non-homogeneous higher order ODEs using the method of variation of parameters.

Outcome 4: Upon completion of this course, students will be able to solve initial value problems using the Laplace transform.

Objectives

1. Demonstrate knowledge of Laplace transforms of basic functions.
2. Demonstrate knowledge inverse Laplace transforms of basic functions.
3. Demonstrate knowledge of utilizing Laplace transforms to solve initial value problems.

Outcome 5: Upon completion of this course, students will be able to solve homogeneous and non-homogeneous systems by the Laplace transform, by using operators, and by using eigenvalues and eigenvectors.

Objectives

1. Demonstrate knowledge of utilizing Laplace transforms to solve systems of ODEs with initial conditions.
2. Demonstrate knowledge of utilizing operator notation to solve systems of ODEs.
3. Demonstrate knowledge of matrix techniques to solve systems of ODEs.

Outcome 6: Upon completion of this course, students will be able to solve differential equations by power series and by Frobenius’ method.

Objectives:

1. Find power series solutions about an ordinary point.
2. Use Frobenius’ method to find series solutions about a regular singular point.

• Communication: The graduate can communicate effectively for the intended purpose and audience.
• Critical Thinking: The graduate can make informed decisions after analyzing information or evidence related to the issue.
• Global Literacy: The graduate can analyze human behavior or experiences through cultural, social, political, or economic perspectives.
• Information Literacy: The graduate can responsibly use information gathered from a variety of formats in order to complete a task.
• Quantitative Reasoning: The graduate can apply quantitative methods or evidence to solve problems or make judgments.
• Scientific Literacy: The graduate can produce or interpret scientific information presented in a variety of formats.

1. First Order Ordinary Differential Equations
   1. Separable ODEs
   2. Exact ODEs
   3. Homogeneous-of-Order-n ODEs
   4. Linear ODEs
   5. Bernoulli ODEs
   6. Applications of First Order ODEs
2. Higher-Ordered ODEs
   1. Reduction of Order
   2. Homogeneous Linear ODEs with Constant Coefficients
   3. Undetermined Coefficients
   4. Variation of Parameters
   5. Cauchy-Euler Equations
   6. Series Solutions at Ordinary Points
   7. Series Solutions at Singular Points
3. Initial Value Problems
   1. Direct solution of IVPs
   2. Laplace Transform and the Solution of an IVP of a Single ODE
   3. Laplace Transform and the Solution of an IVP of a System of ODEs
4. Matrix Methods
   1. Homogeneous Linear Systems with Constant Coefficients
   2. Variation of Parameters for Systems of ODEs
5. Numerical Methods
   1. Eulers Methods
   2. Runge-Kutta 4th Order Method