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# COURSE OUTLINE FOR Differential Equations & Matrix Methods

COURSE DESCRIPTION:

Treatment of ordinary differential equations to include principle types of first and
second order equations; methods of substitution on simple higher order equations;
linear equations and systems of linear equations with constant coefficients;
methods of undetermined coefficients and variation of parameters; Laplace
transforms; series solutions; linear algebra and matrix methods of solutions;
applications to physics and engineering. Prerequisite : MA243

GOALS:

This course is required by the Aerospace Engineering, Electrical Engineering,
Avionics, Engineering Technology and Engineering Physics degree programs. Its
purpose is to provide intermediate mathematical skills for the student to use in
many of the applications he will encounter in future engineering courses.

PERFORMANCE OBJECTIVES:

1. Recognize and solve separable , homogeneous, exact, and linear first order
differential equations.

2. Construct and solve appropriate differential equations for applied problems
involving mixtures, populations, and Newtonian Mechanics.

3. Calculate numerical solutions of differential equations.

4. Solve homogeneous, linear second and higher order differential equations
with constant coefficients.

5. Solve nonhomogeneous, linear second and higher order differential
equations with constant coefficients by the Method of Undetermined
Coefficients and the Method of Variation of Parameters.

6. Construct and solve applied problems involving mechanical vibrations,
forced vibrations, and electric circuits.

7. Compute Laplace transforms of polynomials, exponential and trigonometric
function.

8. Compute inverse Laplace transforms of rational function and solve initialvalue
problems by Laplace Transform Method.

9. Find a power series solution to a given differential equation.

10. Solve a linear system by the Gauss -Jordan elimination method and by Matrix
Methods.

11. Compute eigenvalues and eigenvectors of a given matrix.

12. Solve systems of 1st order linear differential equations by Matrix Methods.

TEXTBOOK:

Nagel, and Saff, Fundamentals of Differential Equations, 4th Edition,

SUGGESTED SUPPLEMENTAL MATERIALS:

a. Robert L. Borrelli and Courtney S. Coleman, Differential Equations,
Modeling Perpectives, John Wiley and Sons, 1997.
b. Beverely West, Steven Strogatz, Jean Marie McDill and John Cantwell,
Interactive Differential Equations, Addison Wesley, 1997.

PERQUISITE KNOWLEDGE BY TOPIC:

1. Properties, derivatives, integrals, and inverses of all the elementary
functions. This includes the trigonometric functions, logarithms and
exponentials
, power functions and hyperbolic functions.

2. Techniques of integration, including integration by parts, variable
substitution, trigonometric substitution, and the method of partial
fractions decomposition .

3. L’Hospital’ s rule .

4. Improper integrals.

5. Partial derivatives.

6. Determinants and Cramer’s Rule.

7. Taylor Series.

8. Vectors, including vector addition, scalar multiplication and dot
product .

 Topics Class Hours Course Objectives 1. First order and simple higher order differential equations. 9 Understand the concepts of a general solution, an initial or boundary condition and the order of a differential equation. Solve equations using techniques such as direct integration, exact differentials, separation of variables, integrating factors and special variable substitutions. 2. Linear omogeneous differential equations -constant coefficients. 8 Understand the concepts of linear operators & linear independence of solutions. Be able to use the Wronskian determinant to test for linear independence. Solve linear homogeneous differential equations in which the solutions of the auxiliary equation are real, complex or repeated . Use Euler’s formula to express complex exponentials in terms of sine and cosine. 3. Nonhomogeneous linear differential equations. 6 Recognize the general form of solutions of nonhomogeneous equations. Use the methods of variation of parameters and either the method of undetermined coefficients or operator methods to construct particular solutions. 4. Applications of linear differential equations. 2 Formulate and solve some simple spring and circuit problems involving linear differential equations with constant coefficients. applications include mechanical resonance. 5. Laplace transforms. 5 Compute Laplace transforms of polynomials, trigonometric and exponential functions. Compute inverse Laplace transforms of rational functions. Use Laplace transforms to solve differential equations. 6. Numerical solutions of differential eqtns. 3 Use linear approximations and power series to solve differential equations. 7. Matrix algebra 3 Multiply matrices by scalars. Add and multiply matrices. Represent systems of linear algebraic equations as matrices and solve by matrix reduction . 8. Matrix inverses 3 Find inverses of matrices using either determinants or matrix reduction. Use matrix inverses to solve systems of linear equations. 9. Eigenvectors and eigenvalues 3 Compute eigenvectors and eigenvalues of a given matrix.
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