Introduces ordinary differential equations by means of numerical, graphical and algebraic analysis. Examines first order differential equations, second and higher order linear equations, methods for nonhomogeneous second order equations, series solutions, systems of first order equations, and Laplace transforms. Covers various applications throughout the course. Requires a graphing calculator with the TI-83/84 Plus recommended.
Goals, Topics, and Objectives
- To provide an introduction to the nature and significance of differential equations for students of engineering, mathematics, and science.
- To demonstrate various applications of differential equations to problems from the physical sciences and engineering.
- To provide methods for solving differential equations that have proved useful in a wide variety of applications.
- To present an exposition of differential equations that incorporates graphical, numerical and algebraic analysis, without undue emphasis on theoretical abstraction or routine mechanical manipulation.
- To use technology to graph solutions of ordinary differential equations (ODEs) and to do explorations and projects involving ODEs.
- Introduction to Differential Equations
- Determine the order of a differential equation and whether it is linear or nonlinear.
- Verify that a function or family of functions solves a differential equation.
- Determine parameter values such that a member of a family of solutions solves a given initial-value problem.
- First-Order Differential Equations
- Relate a solution curve to the direction field for a differential equation.
- Solve a separable differential equation.
- Solve a first-order linear differential equation.
- Solve an exact differential equation.
- Use an appropriate substitution to rewrite and solve a differential equation.
- Apply Euler’s method to obtain a numerical approximation of a differential-equation solution-function value.
- Higher-Order Differential Equations
- Verify that a set of functions is a fundamental set of solutions of a differential equation.
- Use reduction of order to find a second solution of a second-order differential equation given one solution.
- Solve a homogenous linear differential equation with constant coefficients.
- Apply the method of undetermined coefficients to solve a nonhomogeneous linear differential equation with constant coefficients.
- Apply the method of variation of parameters to solve a linear second-order differential equation.
- Solve a Cauchy-Euler equation.
- Solve a system of linear differential equations by elimination.
- Series Solutions of Linear Equations
- Find a Taylor-series solution of an initial-value problem.
- Find power-series solutions of a differential equation about an ordinary point.
- Modeling with Differential Equations
- Solve a problem in the physical sciences (such as a growth or decay problem, a mixture problem, or a Newton’s Law of Cooling problem) whose solution utilizes a first-order linear differential equation.*
- Solve a problem in the physical sciences (such as a spring/mass-system problem) whose solutions utilizes a second-order linear differential equation.*
- Solve a problem in the sciences (such as a logistic-growth problem) whose solutions utilizes a nonlinear differential equation.*
- The Laplace Transform
- Calculate the Laplace transform of a function using the definition of Laplace transform.
- Use algebraic manipulation or a table to find the Laplace transform of a function or its derivatives.
- Use algebraic manipulation or a table to find an inverse Laplace transform.
- Use the Laplace transform to solve an initial-value problem.
- Use the Laplace transform to solve a system of linear differential equations.
Assessment and Requirements
All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives. This exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this exam may be a common exam administered to all sections of Math 289. All additional assessment of student achievement is left to instructor discretion. Some exam problems should require the use of a graphing calculator.
A graphing calculator is required of each student. The Mathematics Division recommends and uses the TI-83/84 Plus Graphing Calculator.
Outcomes
- Mathematics
- Quantitative Literacy
- Category 3: Mathematics (College Algebra Track)