MATH 20063 Elementary Differential Equations PDF

Preview

Summary

P U PInstructional Materials inMATH 20063Elementary Differential Equationscompiled byDMS FacultyCollege of SciencePolytechnic University of the Philippines2020Not For Commercial UseNot For Commercial UseFor the sole noncommercial use of theFaculty of the Department of Mathematics and StatisticsPolyt...

Description

For the sole noncommercial use of the Faculty of the Department of Mathematics and Statistics Polytechnic University of the Philippines 2020

Contributors:

Atienza, Jacky Boy E. Berico, Edwin O. Bernardino, Rhea R. Cabanig, Sarah Jean Q. Costales, Jeffrey A. Pelayo, Sharon Joy F. Poloyapoy, Oscar L. Publico, Juan Jr. L. Torres, Aureluz L. Zablan, Michael G.

Republic of the Philippines POLYTECHNIC UNIVERSITY OF THE PHILIPPINES COLLEGE OF SCIENCE Department of Mathematics and Statistics

Course Title

:

ELEMENTARY DIFFERENTIAL EQUATIONS

Course Code

:

MATH 20063

Course Credit

:

3 UNITS

Pre-Requisite

:

MATH 20053 Calculus 2

Course Description :

This course is an introductory course in ordinary differential equations (ODEs). It focuses primarily on techniques for finding solutions to ODEs. Topics include the existence and uniqueness of solutions, first order ordinary differential equations, linear differential equations, linear equations with constant coefficients, nonhomogeneous equations, undetermined coefficients and variation of parameters, This also involves solving initial value problems using the Laplace transform and its inverse, some applications of fist-order differential equations, and system of first-order differential equations.

COURSE LEARNING PLAN Week Week 1

Week 2

Week 3

Dates

Topics and Subtopics

10/05/20-

• Definitions and Terminology

10-11/20

• Elimination of Arbitrary Constants

10/12/2010/18/20

• Families of Curves • Initial-Value Problems • Direction Fields

10/19/20-

• Separable Variables

10/25/20

• Linear First-Order Equation

Week 4

Week 5

Week 6

Week 7

Week 8

Week 9

Week 10

Week 11

Week 12

Week 13

Week 14

10/26/20-

• Linear First-Order Equation

11/01/20

• Exact Equations

11/02/20-

• Differential Equations with Homogeneous Coefficients

11/08/20 11/09/2011/15/20 11/16/2011/22/20 11/23/2011/29/20 11/30/2012/06/20 12/07/2012/13/20

• Other Methods for First-Order ODE

• Other Methods for First-Order ODE • Homogeneous Linear Equations with Constant Coefficients • Nonhomogeneous Higher Order Differential Equations • Variation of Parameters

12/14/20-

• Definition of Laplace Transforms

12/20/20

• Inverse Transforms

01/04/2101/10/21 01/11/2101/17/21 01/18/2101/24/21

• Solving Linear ODE Using Transforms • Translation Theorems and Additional Operational Properties • Applications of First Order Differential Equations

• System of First-Order Differential Equations

COURSE GRADING SYSTEM The final grade will be based on the weighted average of the student’s scores on each test assigned at the end of each lesson. The final SIS grade equivalent will be based on the following table according to the approved University Student Handbook. Class Standing (CS) = (((Weighted Average of all the Chapter/Unit Tests/Case Study) x 50 )+ 50) Midterm and/or Final Exam (MFE) = (((Weighted Average of the Midterm and/or FinalTests) x 50)+50) Final Grade = (70% x CS) + (30% x MFE) SIS Grade

Final Grade Equivalent

Description

1.00

97.00-100

Excellent

1.25

94.00-96.99

Excellent

1.50

91.00-93.99

Very Good

1.75

88.00-90.99

Very Good

2.00

85.00-87.99

Good

2.25

82.00-84.99

Good

2.50

79.00-81.99

Satisfactory

2.75

77.00-78.99

Satisfactory

3.00

75.00-76.99

Passing

5.00

65.00-74.99

Failure

INC

Incomplete

W

Withdrawn

Prepared by: Rhea R. Bernardino Faculty Member, Department of Mathematics and Statistics College of Science

Contents 1 Introduction to Differential Equations 1.1 Definitions and Terminology…………………………………………….. 1 1.2 Elimination of Arbitrary Constants…………………………………….. 7 1.3 Families of Curves…………………………………………………………. 9 1.4 Initial-Value Problems……………………………………………………. 12 1.5 Direction Fields……………………………………………………………. 15 2 Some Techniques of Solving Ordinary Differential Equations 2.1 Separable Variables………………………………………………………. 17 2.2 Linear First-Order Equation……………………………………………. 24 2.3 Exact Equations…………………………………………………………… 33 2.4 Differential Equations with Homogeneous Coefficients……….……. 38 2.5 Other Methods for First-Order ODE…………………………………… 42 3 Higher-Order Differential Equations 3.1 Homogeneous Linear Equations with Constant Coefficients………. 59 3.2 Nonhomogeneous Higher Order Differential Equations……………. 68 3.3 Variation of Parameters………………………………………………….. 76 4 The Laplace Transforms 4.1 Definition of Laplace Transforms……………………………………….. 83 4.2 Inverse Transforms………………………………………………………… 88 4.3 Solving Linear ODE Using Transforms………………………………… 90 4.4 Translation Theorems and Additional Operational Properties…….. 93 5 Applications of First Order Differential Equations 5.1 Differential Equations of Plane Curves……………………………….... 101 5.2 Isogonal and Orthogonal Trajectories………………………………...… 104 5.3 Newton’s Law of Cooling (and Heating) ……………………………..… 109 5.4 Exponential Law of Growth and Decay………………………………… 114 .5 Simple Electric Circuits…………………………………………………..… 123 5.6 Mixture Problems………………………………………………………….. 129 5.7 Velocity of Escape from the Earth………………….…………………… 134 5.8 Newton’s Second Law……………………………………………………… 137 5.9 Flow of Water Through an Orifice……………………………….……… 138 6 System of First-Order Differential Equations 6.1 System of Differential Equations……………………………………….. 139 6.2 Homogeneous Linear Systems with Constant Coefficients………… 146 6.3 Matrix Exponential………………………………………………………… 153

Unit Test No. 1 Classifications of Differential Equations, Elimination of Arbitrary Constants, Families of Curves, Separation of Variables, Reducible to Separation of Variables GENERAL INSTRUCTIONS 1. Work INDEPENDENTLY and HONESTLY. 2. Use permanent black or blue-inked pens only. 3. Write the questions, your answers and solutions in short white bond paper (8.5” × 11”). 4. Write your FULL NAME, COURSE, YEAR, SECTION on the upper left part of each page of your answer sheets. A. COMPLETE THE TABLE. For each given differential equation, determine the dependent variable(s) (DV) and independent variable(s) (IV), classify according to type (write only ODE for ordinary differential equation and PDE for partial differential equation), order and linearity (L for linear and NL for nonlinear differential equation) and give its degree.

(1) (2) (3) (4) (5) (6) (7) (8)

Differential Equation dy = 4x2 y − y 2 dx 4xy′′ + 5y ′ − y sin x = 0 s   4 2 3 dy dy + 3y = dx dx4 ∂y ∂w =0 + ∂x ∂x y(y ′ )4 − 5(y ′′ )2 + 2 = 0 s  3 dy d2 y = 5+ 2 dx dx (2x − y)dx + (2x − 3y)dy = 0 sin θ

d3 y dy − cos θ 3 dθ dθ

DV

IV

Type Order

Degree Linearity

B. SOLVING. Answer each of the following. Write your solution as neatly as possible. Simplify your final answers by (a) converting logarithmic expressions into single logarithm, if there are any (b) simplifying complex fractions. 1. By determinants, eliminate the arbitrary constants c1 and c2 : y = c1 e2x + c2 ex +

1 2

2. Eliminate the arbitrary constants c1 and c2 : y = c1 ex sin x + c2 ex cos x 3. Find the differential equation of the family of ellipses with center at the origin a the and major axis on the y − axis. 4. Find the differential equation of the family circles entered at (h, k) passing through the origin and (0, 4). 5. Solve: (e−y + 1)−2 ex dx + (e−x + 1)−3 ey dy = 0 6. Solve: sin x cos2 y dx + cos2 x dy = 0, y(0) =

π 4

7. Solve: (2x − 2y − 1)dx − (x − y + 1)dy = 0 8. Find a continuous solution satisfying   x if 0 ≤ x < 1

dy + 2xy = f (x) where f (x) =  0 dx

if x ≥ 1

, y(0) = 2

Unit Test No. 2 Techniques of Solving Differential Equations GENERAL INSTRUCTIONS 1. Work INDEPENDENTLY and HONESTLY. 2. Use permanent black or blue-inked pens only. 3. Write the questions, your answers and solutions in short white bond paper (8.5” × 11”). 4. Write your FULL NAME, COURSE, YEAR, SECTION on the upper left part of each page of your answer sheets. A. (Linear Differential Equations) Find the general (or particular) solution of the given differential equations and give the largest interval over which the general solution is defined. 1. cos2 x sin x dy + (y cos3 x − 1)dx = 0   dy π 2. sin x + y cos x = x, y =2 dx 2 B. (Exact Equations) Find the general (or particular) solution of the given differential equations.     1. ex + 2xey + x2 dx + x2 ey + y 2 dy = 0 2. (4xy + 3x2 )dx + (2y + 2x2 )dy = 0, y(0) = −2

C. (Differential Equations with Homogenous Coefficients) Find the general (or particular) solution of the given differential equations.   1. 6x2 − 7y 2 dx − 14xydy = 0   2. x + yey/x dx − xey/x dy = 0, y(1) = 0 D. (Bernoulli Differential Equation) Find the general (or particular) solution of the given differential equations. 1.

dx x + = −y 5 x9 dy y

2. x2

1 dy − 2xy = 3y 4 , y(1) = dx 2

Unit Test No. 3 Techniques of Solving Differential Equations, Higher-Order Homogeneous Linear Equations with Constant Coefficients GENERAL INSTRUCTIONS 1. Work INDEPENDENTLY and HONESTLY. 2. Use permanent black or blue-inked pens only. 3. Write the questions, your answers and solutions in short white bond paper (8.5” × 11”). 4. Write your FULL NAME, COURSE, YEAR, SECTION on the upper left part of each page of your answer sheets. A. Solve the following differential equations by using appropriate method or substitutions. 1.

√ dy = 2 + y − 2x + 3 dx

2. y ′ − (4x − y1 )2 = 0 3. (x + y − 1)dx + (y − x − 5)dy = 0 4. 6xydx + (4y + 9x2 )dy = 0 B. Determine the general solution for each given higher order differential equation. Write your solutions as neatly as possible. Show computations for the roots of the auxiliary equation. 1. y ′′′ − 2y ′′ − 15y ′ = 0 2. y ′′ + 4y ′ + 29y = 0 3. y ′′′ − 5y ′′ + 3y ′ + 9y = 0 d2 y d4 y + 24 + 9y = 0 dx2 dx4  3  2  3 5. D2 + 9 D2 − 4D + 4 D2 − 7D − 18 y = 0

4. 16

Unit Test No. 4 Higher-Order Nonhomogeneous Linear Equations with Constant Coefficients, Laplace and Inverse Laplace Transforms GENERAL INSTRUCTIONS 1. Work INDEPENDENTLY and HONESTLY. 2. Use permanent black or blue-inked pens only. 3. Write the questions, your answers and solutions in short white bond paper (8.5” × 11”). 4. Write your FULL NAME, COURSE, YEAR, SECTION on the upper left part of each page of your answer sheets. A. Find a linear differential operator that annihilates the given function. 1. 13x + 9x2 − sin 4x 2. 3 + ex cos 2x B. Determine the form of a particular solution for each differential equation. 1. y ′′ − 2y ′ + y = 10e−2x cos x 2. y ′′′ − 4y ′′ + 4y ′ = 5x2 − 6x + 4xe2x + 3e5x

C. Solve the given differential equation by undetermined coefficients. 1. y ′′ + 6y ′ + 9y = −xe4x 2. y (4) − 2y ′′′ + y ′′ = ex + 1

D. Solve the given differential equation by variations of parameters. 1. y ′′ + y = cos2 x 2. y ′′ + 2y ′ + y = e−t ln t E. Evaluate the following. 1. L{e3t + cos 6t − e3t cos 6t} 2. L{3 sinh 2t + 3 sin 2t}   3 6s −1 + 3. L s2 + 25 s2 + 25   3s − 2 −1 4. L 2s2 − 6s − 2   se−4s −1 5. L (3s + 2)(s − 2)

F. Solve: y ′′ − 6y ′ + 15y = 2 sin 3t, y(0) = −1, y ′ (0) = −4

Final Exam GENERAL INSTRUCTIONS 1. Work INDEPENDENTLY and HONESTLY. 2. Use permanent black or blue-inked pens only. 3. Write the solutions (if necessary) and letter of choice in short white bond paper (8.5” × 11”). If your answer is not among the choices, choose letter E. 4. Write your FULL NAME, COURSE, YEAR, SECTION on the upper left part of each page of your answer sheets.

Multiple Choice. 1. Determine the order and degree of the differential equation 2

−2x



d4 y dx4

2

 2 4 dy − 1 = 0. + 4x dx2

(a) order: 2, degree:4

(b) order: 4, degree:2

(c) order: 2, degree:2

(d) order: 4, degree:4

2. The equation y 2 = cx is the general solution of which of the following differential equation? dy 2y 2y dy y dy dy y (a) = =− (b) = (c) (d) =− 2x dx dx x dx 2x x dx 3. Find the equation of the curve at every point of which the tangent line has a slope of 4x. (a) y = −2x2 + C

(b) x = 2y 2 + C

(c) x = −2y 2 + C

(d) y = 2x2 + C

4. Which of the following represents the differential equation of the family of parabolas having their vertices at the origin and their foci on the negative x−axis? (a) 2xdy + ydx = 0 (b) xdy + ydx = 0 (c) 2xdy − ydx = 0 (d) 2ydx − xdy = 0 5. Which of the following equations is not a variable separable? xy + 3x − y − 3 dy dy = = e3x+2y (b) (a) xy − 2x + 4y − 8 dx dx  2  dy xy + 2y − x − 2 (c) 2ydx = x + 2xy + y 2 dx (d) = xy − 3y + x − 3 dx

6. Which of the following equations is an exact differential equation?   (a) x2 + 1 dx − xydy = 0 (b) xdy + (3x − 2y)dx = 0   (c) 2xdy + 2 + x2 dy = 0 (d) x2 ydy − ydx = 0

7. Which of the following is a differential equation with homogeneous coefficient?   p (a) y 2 − x2 + y 2 dx − x2 dy = 0 (b) (x + 2y − 1)dx = (x + 2y − 3)dy   p 2 2 (c) y − x + y dx − xdy = 0        y y y dy − tan + 1 dx = 0 (d) tan x x x

8. Determine the general solution of the the differential equation: (x + y)dy = (x − y)dx (a) x2 − 2xy + y 2 = c

(b) x2 + 2xy + y 2 = c

(c) x2 − 2xy − y 2 = c

(d) x2 + y 2 = c

For the next five questions, consider the following differential equation:   (y 2 + xy3 )dx + 5y 2 − xy + y 3 sin y dy = 0

Let M = y 2 + xy3 and N = 5y 2 − xy + y 3 sin y 9.

∂M =? ∂y (a) 2y + 3xy2

10.

(b) 2y − 3xy 2

(c) y 2 x + y 3

(d) y 2 x − y 3

(b) −y

(c) 5y 2 + y 3 sin y

(d) 5y 2 − y + y 3 sin y

(c) 3y + 3xy 2

(d) y + 3xy2

(c) 3y

(d) −3y

∂N =? ∂x

(a) y   ∂N ∂M =? 11. − ∂x ∂y

(a) 3y − 3xy2 (b) y − 3xy 2   1 ∂M ∂N 12. =? − M ∂y ∂x 3 3 (b) − (a) y y

Page 2

13. The integrating factor for the given differential equation is 1 (a) y 3 (b) 3 (c) 3y y

(d) −3y

14. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96mg. How many mg will be left after 200 years? (a) 88.60

(b) 95.32

(c) 92.16

(d) 90.72

15. Radium decomposes at a rate proportional to the amount present. If half of the original amount disappears after 1000 years, what is the (approximate) percentage lost in 100 years? (a) 6.70%

(b) 4.50%

(c) 5.36%

(d) 4.30%

16. According to Newton?s Law of Cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is 30o and the substance cools from 100o to 70o in 15 minutes, approximately, how long will it take to cool 100o to 50o ? (a) 33.58 minutes

(b) 43.50 minutes

(c) 35.39 minutes

(d) 45.30 minutes

17. Determine the equation of the family of orthogonal trajectories of the the family of curves defined by y 2 = 2x + c (a) y = ce−x

(b) y = ce2x

(c) y = ce−2x

(d) y = cex

18. Determine the roots of the auxiliary equation of y ′′′ + 3y ′′ − 4y ′ − 12y = 0 (a) −3, −2, 2

(b) −3, −2, 3

(c) 0, 2, −2

(d) −3, 2, 3

(c) 2x

(d) 2

19. Evaluate W (1 + x, x, x2 ) (a) −2

(b) −2x

20. Which of the following best represents the particular solution of the differential equation: y ′′ − 6y ′ + 9y = 6x2 + 2 − 12e3x ? (a) yp = Ax2 + Bx + C + Ee3x

(b) yp = Ax2 + Bx + C + Ex2 e3x

(c) yp = Ax2 + Bx + C + Exe3x

(d) yp = Ax2 + Bx + C + Ex3 e3x

21. The complementary solution, yc, of the equation y ′′′ + 2y ′′ + y ′ = 10 is (a) yc = c1 e−x + c2 e−x + c3

(b) yc = c1 ex + c2 xex + c3

(c) yc = c1 e−x + c2 ex + c3

(d) yc = c1 e−x + c2 xe−x + c3 Page 3

22. Solve: y ′′′ + 2y ′′ + y ′ = 10 (a) y = c1 + c2 e−x + c3 xe−x + 10x2

(b) y = c1 + c2 e−x + c3 xe−x + 10x

(c) y = c1 x + c2 e−x + c3 xe−x + 10x   4s 23. Evaluate L 4s2 + 1 t t (a) cos (b) sin 2 2

(d) y = c1 + c2 e−x + c3 xe−x + 10

(c) cos 2t

24. Evaluate L{te2t sin 6t} 12s − 24 12s − 24 12s − 24 (b) (c) (a) 2 2 2 [(s − 2) + 36] [(s − 2) + 36] [(s − 2)2 + 6]2   2s + 5 −1 25. Evaluate L (s − 3)2 (a) e−t + e−3t

(b) 2e−t + 11e−3t

(c) 2e−t + 11te−3t

Page 4

(d) sin 2t

(d)

12s − 24 [(s + 2)2 + 36]2

(d) e−t + 11te−3t

1

Chapter 1: Introduction to Differential Equations Chapter Overview This chapter focused on the discussion of some definitions and concepts in differential equations as well as its classifications. Differential equations representing families of curves and an introduction to initial value problem are also included in this chapter. Learning Outcomes At the end of this chapter, the student is expected to 1. define a differential equation 2. classify a differential equation according to its type, order, degree and linearity 3. obtain a differential equation by eliminating arbitrary constants

1.1 Definitions and Terminology Definition 1: Differential Equation An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE). A differential equation can be classified according to its type, order, and linearity.

I. Classification by Type An equation involving only ordinary derivatives of one or more dependent variables with respect to a single independent variable it is said to be an ordinary differential equation (ODE). Example 1. dy + 5y = e2x , dx equations.

d2 y dx dy d3 y − + 5y = 0, and + = ex + y are ordinary differential 2 3 dx dt dx dt

An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a partial differential equation (PDE). Example 2. ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂u ∂u ∂v + = 0, = 2 − 2 , and are partial differential equations. =− 2 2 2 ∂x ∂t ∂t ∂y ∂x ∂y ∂x

2 Notations:

dn y dy d2 y d3 y , , , . . . , ; dxn dx dx2 dx3 to denote the order of the derivative.

Throughout the discussion, we will use the Leibniz notation or the prime notation y ′ , y ′′ , y′′′ , y (4), y (5) , . . . , y(n)

II. Classification by Order The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. Example 3. d3 y d2 y − 2 + 5y = 0 is a third-order ordinary differential equation 1. dx dx3 ∂ 2u ∂ 2u ∂u is a second-order partial differe...