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June 19th

June 19th

# Differential Equations

Department: Mathematics
Credit Hours: 3
Prerequisite: MTH 235
General Education: Quantitative Competence
College Learning Outcomes: 9a, 9b, 9c, 9d, 9e – Quantitative Competence and
2b – Critical Thinking

I. Course Description: This course studies methods for solving ordinary differential equations
of first second and higher order. It includes applications, series, systems and numerical techniques .

II. Purpose of the Course: Differential equations are an excellent vehicle for displaying the
interrelations between mathematics and the physical sciences. The student can see ways in
which the solutions to specific problems have benefited from work of a more abstract nature.

III. III. College Learning Outcomes and Objectives: (This course is de signed to help students
fulfill the Quantitative Competence Learning Outcome through achievement of the fol lowing
learning objectives:

(9a) Students can formulate specific questions from vague problems, select effective
problem- solving strategies , and know which mathematical operations are appropriate in
particular contexts.

(9b) Students can perform mental calculations and estimates with proficiency, and decide
when an exact answer is needed and when an estimate is more appropriate.

(9c) Students can use a calculator correctly, confidently, and appropriately and/or use

(9d) Students can use tables, graphs, s preadsheets and statistical techniques to organize,
interpret and present numerical information.

(9e) Students can judge the validity of quantitative results presented by others.

(2b) Students can demonstrate the ability to reflect on issues and/or theories systematically.

IV. Course Objectives:

1. Students should be able to solve first order equations .(LO 9a-e, 2b)
2. Students should be able to solve second order linear equations .(LO 9a-e)
3. Students should be able to apply series solutions. (LO 9a-e, 2b)
4. Students should be able to solve third and higher order linear equations. (LO 9a-e)
5. Students should be able to use the LaplaceTransform. .(LO 9a-e, 2b)
6. Students should be able to solve systems of linear equations. .(LO 9a-e)
7. Students should be able to use numerical methods to solve differential equations. .(LO 9a-e, 2b)

V. Topical Outline

A. First Order Differential Equations
1. Linear equations
2. Separable equations
3. Non-linear equations

B. Application of first order linear equations
1. Population dynamics and some related problems
2. Problems in mechanics
3. Exact equations and integrating factors
4. Homogeneous equations
5. Miscellaneous problems and applications
6. First order difference equations

C. Second order linear equations
1. Homogeneous equations with constant coefficients
2. Fundamental solutions of linear homogeneous equations
3. Linear independence and the Wronskian
4. Complex roots of the characteristic equation
5. Repeated roots
6. Nonhomogeneous equations; method of unde termined coefficients
7. Variation of parameters
8. Mechanical and electrical vibrations
9. Forced vibrations

D. Series solutions of second order linear equations
1. Power series
2. Series solutions near an ordinary point
3. Regular singular points
4. Euler equations

E. Higher order linear equations
1. Homogeneous equations with constant coefficients
2. The method of undetermined coefficients
3. The method of variation of parameters

F. The Laplace Transform
1. Definition of the Laplace Transform
2. Solution of initial value problems
3. Step function
4. Impulse function
5. The convolution integral

G. Numerical Methods
1. The Euler method
2. Errors in numerical procedures
3. The Runga-Kutta method
4. Systems of first order equations

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