MAT-274 Differential Equations
This course is an introductory survey of ordinary differential equations. First order differential equations and methods for obtaining solutions are investigated. Methods include integration, variation of parameters, and integrating factors. These methods are generalized for second order differential equations. Additional methods include numerical approximation, Laplace Transforms, and power series.
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Course Outline
Course Description
This course is an introductory survey of ordinary differential equations. First order differential equations and methods for obtaining solutions are investigated. Methods include integration, variation of parameters, and integrating factors. These methods are generalized for second order differential equations. Additional methods include numerical approximation, Laplace Transforms, and power series.
Credit Hours
4Semester(s) Course Will Be Offered
Fall, Spring, Summer
Course Learning Outcomes
- Use the language and notation of elementary differential equations
- Recognize and/or classify various types of elementary differential equations
- Observe an application situation and apply the appropriate differential model
- Explain the concepts behind various transformations used in differential equations
- Interpret graphically the behavior of solutions to differential equations
- Evaluate their results for reasonableness
Topic Outline
- 1. Introduction to ordinary differential equations
- a. Terminology and notation
- b. Initial value problems (IVPs)
- c. Solutions to differential equations and IVPs
- 2. Applications of differential equations and IVPs (topics may vary at the instructor's discretion)
- 3. First-order differential equations
- a. Existence and uniqueness theorem
- b. General and particular solutions
- c. Techniques for solving:
- i. Integration
- ii. Variation of parameters
- iii. Integrating factors
- iv. Transformation
- d. Solving differential equations that are:
- i. Solvable by integration (y′ = 𝑓(x))
- ii. Linear
- iii. Separable
- iv. Bernoulli
- v. Homogeneous
- vi. Exact
- vii. Transformable into one of the types listed above
- e. Numerical approximation of solutions:
- i. Euler's method
- ii. Improved Euler's method
- iii. Runge-Kutta method
- 4. Second-order differential equations
- a. General and particular solutions
- b. Fundamental set of solutions
- i. Linear independence of solutions
- ii. The Wronskian
- c. Types investigated:
- i. Autonomous
- ii. Linear with constant coefficients
- (1) Real-valued solutions to homogeneous equations
- (a) The characteristic polynomial
- (b) Solutions when the characteristic polynomial has two real roots
- (c) Solutions when the characteristic polynomial has only one real root
- (d) Real-valued solutions when the characteristic polynomial has complex roots
- (2) Finding particular solutions by the method of undetermined coefficients
- iii. Equations where one solution to the complementary equation is known and reduction of order can be used to obtain the general solution
- iv. Equations where a fundamental set of solutions is known for the complementary equation and variation of parameters can be used to obtain the general solution
- v. (Cauchy-)Euler equation
- vi. Linear with polynomial coefficients (series solution)
- d. Power series method (introduction; scope may vary at the instructor's discretion)
- i. Review of power series
- ii. Power series solutions
- iii. Ordinary and singular points
- e. Laplace transforms (introduction; scope may vary at the instructor's discretion)
- i. General properties
- ii. Evaluation of transforms
- iii. Inverse transforms
- iv. Solving differential equations using Laplace transforms
- f. Fourier series (optional)
- i. General properties
- ii. Evaluating coefficients of the series
- iii. Solving differential equations using Fourier series