# MATH 2320

# Differential Equations

### Math 2320

**State Approval Code:**27030015119**Semester Credit Hours:**3**Lecture Hours per Week:**3**Contact Hours per Semester:**48

### Catalog Description

Study of ordinary differential equations including linear equations, systems of equations, series solutions Laplace transforms; includes introduction to boundary value problems, uniqueness of solutions, singular points, partial differential equations and Fourier series. (27030015119) (3-3-0) (spring only)

### Prerequisites

Prerequisite: Math 2414 Corequisite: Math 2415

### Course Curriculum

### Basic Intellectual Compentencies in the Core Curriculum

- Reading
- Writing
- Speaking
- Listening
- Critical thinking
- Computer literacy

### Perspectives in the Core Curriculum

- Establish broad and multiple perspectives on the individual in relationship to the larger society and world in which he/she lives, and to understand the responsibilities of living in a culturally and ethnically diversified world.
- Stimulate a capacity to discuss and reflect upon individual, political, economic, and social aspects of life in order to understand ways in which to be a responsible member of society.
- Recognize the importance of maintaining health and wellness.
- Develop a capacity to use knowledge of how technology and science affect their lives.
- Develop personal values for ethical behavior.
- Develop the ability to make aesthetic judgments.
- Use logical reasoning in problem solving.
- Integrate knowledge and understand the interrelationships of the scholarly disciplines.

### Core Components and Related Exemplary Educational Objectives

### Communication (composition, speech, modern language)

- To participate effectively in groups with emphasis on listening, critical and reflective thinking, and responding.

### Mathematics

- To apply arithmetic, algebraic, geometric, higher-order thinking, and statistical methods to modeling and solving real-world situations.
- To represent and evaluate basic mathematical information verbally, numerically, graphically, and symbolically.
- To expand mathematical reasoning skills and formal logic to develop convincing mathematical arguments.
- To use appropriate technology to enhance mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of the results.
- To interpret mathematical models such as formulas, graphs, tables and schematics, and draw inferences from them.
- To recognize the limitations of mathematical and statistical models.
- To develop the view that mathematics is an evolving discipline, interrelated with human culture, and understand its connections to other disciplines.

### Instructional Goals and Purposes

Panola College's instructional goals include 1) creating an academic atmosphere in which students may develop their intellects and skills and 2) providing courses so students may receive a certificate/an associate degree or transfer to a senior institution that offers baccalaureate degrees.

### General Course Objectives

Successful completion of this course will promote the general student learning outcomes listed below. The student will be able

1. To apply problem-solving skills through solving application problems.

2. To demonstrate arithmetic and algebraic manipulation skills.

3. To read and understand scientific and mathematical literature by utilizing proper vocabulary and methodology.

4. To construct appropriate mathematical models to solve applications.

5. To interpret and apply mathematical concepts.

6. To use multiple approaches - physical, symbolic, graphical, and verbal - to solve application problems

### Specific Course Objectives

Major Learning Objectives Essential Competencies

Upon completion of **MATH 2320**, the student will be able to demonstrate:

1. Competence in classifying differential equations as to ordinary, partial, linear, non-linear, order and degree, and to construct differential equations under given conditions.

2. Competence in solving first order differential equations employing the techniques of variables separable, homogeneous coefficient, or exact equations.

3. Competence in solving applied problems which are linear in form.

4. Competence in solving linear differential equations employing the techniques of integrating factors, substitution, variation of parameters and reduction of order.

5. Competence in finding the Laplace Transform of specified functions and solving linear ordinary differential equations using the Laplace Transform.

### General Description of Each Lecture or Discussion

After studying the material presented in the text(s), lecture, laboratory, computer tutorials, and other resources, the student should be able to complete all behavioral/learning objectives listed below with a minimum competency of 70%.

**Definitions, Families of Curves**

Upon completion of this section, the student will be able to correctly

1.1 Identify a differential equation (DE).

1.2 Identify the independent variable in a DE. 1.3 Identify the dependent variable in a DE.

2.1 Determine the order of a DE.

2.2 Classify a DE as linear or nonlinear.

2.3 Classify a DE as an ODE or PDE.

3.1 Eliminate the arbitrary constants from a given relation by the generation of an appropriate DE.

4.1 Obtain the DE of a specified family of plane curves and sketch the representative
members of the family. **Equations of Order One **Upon completion of this section, the student will be able to correctly

5.1 Optional:

6.1 State the conditions under which a DE has a unique solution.

7.1 Identify and classify, by inspection, an ODE as being separable.

7.2 Obtain the general solution of a separable ODE.

8.1 Identify and classify, by inspection, homogeneous functions.

8.2 Identify the degree of a homogeneous function.

8.3 State the formal definition of homogeneity.

9.1 Identify and classify ODE's with homogeneous coefficients.

9.2 Solve ODE's with homogeneous coefficients by using the appropriate substitution **y = vx **or **x = vy **whichever leads to the more efficient solution.

10.1 State the definition of an exact ODE.

10.2 Test an ODE for exactness.

10.3 Solve an exact ODE.

11.1 State the definition of a first order linear differential equation.

11.2 Write a first order linear ODE in standard form.

11.3 Determine the integrating factor (**IF**) of a first order linear differential equation.

11.4 Solve a first order linear ODE by application of the **IF**.

12.1 Solve for the general solution of a linear ODE.

**Elementary Applications **Upon completion of this section, the student will be able to correctlyFormulate the
ODE and then solve the resulting equation for *at least *the following types of elementary applications:

(1) Growth and decay (i) Population growth (ii) Half-life

(2) Newton's Law of Cooling

(3) An L-R series circuit

(4) A mixture problem

(5) Falling body problems

(6) Frictional forces

(7) Velocity of Escape

(8) Chemical Conversions (Law of Mass Action)

(9) Rates of Dissolving

(10) Logistic Growth and the Price of Commodities

(11) Orthogonal trajectories

Applications of Derivatives Upon completion of this section, the student will be able to correctly

18.1 Identify and state from memory the exact differentials given in the textbook.

18.2 Identify and state from the memory the exact differentials of (a) d{ln(y/x)] (b) d[ln(xy)] (c) d[(1/2)ln(x2 + y2)]

18.3 Solve ODE's using the above referenced lists of differentials.

18.4 Solve ODE's using an IF of the form xkyn to make the given ODE exact.

19.1 Determine the IF (integrating factor) for a first order ODE.

19.2 Apply IF's to convert first order ODE's into exact equations and then solve.

20.1 Solve an ODE of the form**M dx + N dy = 0**by the use of a suitable substitution as suggested by the equation.

21.1 State the general form of Bernoulli's equation.

21.2 State the conditions under which a Bernoulli equation is linear and hence solvable as such.

21.3 Solve Bernoulli equations using the transformation v = y1 - n or z = x1 - n as appropriate.

22.1 Solve ODE's with coefficients linear in two variable using a transformation of
the form **w = ax + by**.

23.1 *Classify and then solve *any of the ODE's in the Miscellaneous Exercise. **Linear Differential Equations **Upon completion of this section, the student will be able to correctly 24.1 State
the general form of a linear ODE of order n.

24.2 Distinguish between a linear ODE of order n that is homogeneous and one that is nonhomogeneous.

24.3 State the Superposition Principle.

25.1 State Theorem 5 of the textbook.

26.1 State the definition of linear independence.

26.2 State the definition of linear dependence.

27.1 State the definition of the Wronskain.

27.2 State the relationship**s **between the vanishing of the Wronskian and dependence and independence.

28.1 State and prove Theorem 7 of the text.

29.1 State the meaning of the phrase "general solution of a nonhomogeneous equation." 30.1 State the definition of the differential operator Dk.

31.1 State the fundamental laws of operators with both constant and variable coefficients (primary emphasis being given to constant coefficient operators).

32.1 State and use the exponential shift property of differential operators. **Linear Equations with Constant Coefficients **Upon completion of this section, the student will be able to correctly

33.1 Optional.

34.1 Formulate the auxiliary equation of a linear ODE with constant coefficients. 34.2 Use the auxiliary equation to solve a linear ODE with constant coefficients and distinct roots. 35.1 Use the auxiliary equation to solve a linear ODE with constant coefficients and repeated roots or both distinct and repeated roots.

36.1 State Euler's Formulae.

37.1 Use the auxiliary equation to solve a linear ODE with constant coefficients where the auxiliary equation has complex roots and any combination of distinct, repeated, and complex roots.

38.1 State the definitions of sinh(x) and cosh(x).

38.2 State the basic properties and identities of the hyperbolic functions. **Nonhomogeneous Equations: Undetermined Coefficients **Upon completion of this section, the student will be able to correctly

39.1 Construct a linear ODE with real, constant coefficients that is satisfied by a given function.

40.1 Optional.

41.1 Obtain the general solution of an ODE by the **Method of Undetermined Coefficients **as outlined in Steps (a) through (d) on page 126 of the text.

41.2. State and apply the statement: The Method of Undetermined Coefficients is applicable only to those ODE's for which R(x) is a solution of a homogeneous linear equation with constant coefficients.

42.1 Obtain a particular solution of a nonhomogeneous equation by inspection for the
following cases: (a) R(x) = R0, R0 a constant and bn = 0 (b) R(x) = R0, and Dky the
lowest ordered derivative that actually appears in the ODE. (c) R(x) = Asin(Bx) and
R(x) = A cos(Bx) (d) R(x) = eax (e) R(x) is any linear combination of the above functions. **Variation of Parameters **Upon completion of this section, the student will be able to correctly

43.1 Optional.

44.1 Solve a nonhomogeneous linear equation with constant coefficients by reduction of order.

45.1 Obtain the solution of a nonhomogeneous linear equation by **Variation of Parameters. **46.1 Solve the equation y'' + y = f(x) for unspecified f(x).

46.2 Solve the equation y'' - y = f(x) for unspecified f(x).

46.3 Solve the ODE's in the Miscellaneous Exercise, p. 151. **Inverse Differential Operators**Upon completion of this section, the student will be able to correctly

47.1 Use exponential shift to find a particular and general solutions to a particular ODE.

48.1 Optional. 49.1 Evaluate [1/f(D)]eax and apply to find the particular and general solutions of a given ODE. 49.2 Evaluate [D2 + a2] -1 sin(ax) and [D2 + a2] -1 cos(ax) and apply to find the particular and general solutions of a given ODE.

**The Laplace Transform **Upon completion of this section, the student will be able to correctly

60.1 State some elementary transformations including (i) differentiation and (ii) integration. 60.2 State the definition of a linear transformation.

60.3 State the definition of a general linear integral transformation.

60.4 Define the term kernel as applied to transforms.

61.1 State the complete definition of the Laplace transformation.

61.2 Prove that the Laplace transform is a linear transformation.

61.3 State and prove **The First Translation Theorem **(Theorem A of Notes).

61.4 State and prove **The Change of Scale Theorem **(Theorem B of Notes).

62.1 Derive the Laplace transforms of at least the following elementary functions: (a) f(t) = eat (b) f(t) = 1 (c) f(t) = sin (kt) (d) f(t) = cos (kt) (e) f(t) = tn, for n a positive integer. (f) f(t) = sinh (kt) (g) f(t) = cosh (kt)

62.2 State from memory the transform of each of the functions in Objective

62.1 above.

63.1 State the definition of a **sectionally continuous **(*piecewise continuous*) function.

64.1 State the definition of exponential order.

64.2 Prove that a given function is (or is not) of exponential order.

64.3 State and prove Theorem 8 on p. 193 of the text.

64.4 State and prove Theorem 9 on p. 193 of the text.

65.1 State the definition of a function of Class A.

65.2 State the **sufficient conditions **for the Laplace transform of a function to exist.

65.3 State Theorem 10 on p. 195 of the textbook. 65.4 State Theorem 11 on p. 196 of
the textbook.Ex.1 State the definition of the **Unit Step Function **(c.f., Section 73 of textbook). Ex.2 State and prove Theorem C at the end of this
list of Objectives.

66.1 State and prove Theorem 12 on p. 198 of the textbook.

66.2 State and prove Theorem 13 on p. 198 of the textbook.

66.3 State and prove Theorem 14 on p. 200 of the textbook.

67.1 State and prove Theorem 15 on p. 200 of the textbook.

67.2 State and prove Theorem 16 on p. 200 of the textbook.

67.3 Use the above Theorems to find Laplace transforms.

68.1 State the definition of the Gamma Function.

68.2 State and prove Theorem 17 on p. 202 of the textbook.

68.3 State and prove Theorem 18 on p. 202 of the textbook.

68.4 Evaluate:

68.5 Evaluate: Ex.3 State and prove Theorem D at the end of this list of Objectives. Ex.4 Use Theorem D to find Laplace transforms. Ex.5 State Theorem E at the end of this list of objectives.

69.1 Optional. **Inverse Transforms **Upon completion of this section, the student will be able to correctly

70.1 State the definition of the inverse Laplace transform.

70.2 Write from memory *at least *the following inverse Laplace transforms: (a) (b) (c) (d) (e) (f) 70.3 State the conditions
under which the inverse Laplace transform exists.

70.4 State Theorem 20 on p. 210 in the textbook.

70.5 State Theorem 21 on p. 211 in the textbook.

70.6 Find the inverse Laplace transform for a F(s).

71.1 Use partial fraction expansions to find inverse Laplace transforms. Ex.6 Use Theorem A and the concept of the inverse Laplace transform to find f(t) given F(s). Ex.7 Use Theorem C and the concept of the inverse Laplace transform to find f(t) given F(s). Ex.8 Use Theorem E (The Convolution Theorem, Section 73 of text) and the concept of the inverse Laplace transform to find f(t) given F(s).

72.1 Use the concept of the inverse Laplace transform to solve initial value problems that are linear ODE's with constant coefficients.

Methods of Instruction/Course Format/DeliveryMethods employed will include Lecture/demonstration, discussion, problem solving,
analysis, and reading assignments. *Homework will be assigned. *Faculty may choose from, but are not limited to, the following methods of instruction:

(1) Lecture

(2) Discussion

(3) Internet

(4) Video

(5) Television

(6) Demonstrations

(7) Field trips

(8) Collaboration

(9) Readings

### Assessment

Faculty may assign both in- and out-of-class activities to evaluate students' knowledge and abilities. Faculty may choose from – but are not limited to -- the following methods

Attendance

Book reviews

Class preparedness and participation

Collaborative learning projects

Compositions

Exams/tests/quizzes

Homework

Internet

Journals

Library assignments

Readings

Research papers

Scientific observations

Student-teacher conferences

Written assignments

Four (4) **Major Exams **at 15% each **60% Homework Notebook/Folder 10% Note: There will be no make-up exams**. If you miss an exam your Final Exam percentage will be used as a substitute for
the missing grade. If you **do not miss **any exams, your one lowest Exam grade will be replaced by the Final Exam percentage
provided it (the Final Exam percentage) is higher. **Comprehensive Final Examination 30% Letter Grades for the Course will be assigned
as follows: A: **90 <**Average **< 100 **B: **80 < **Average **< 90 **C: **70 < **Average **< 80 **D: **60 < **Average **< 70 **F: **00 <**Average **< 60

### Text, Required Readings, Materials, and Supplies

For current texts and materials, use the following link to access bookstore listings.