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84 Pages

### Chapter 5

Course: MATH 3321, Fall 2010
School: U. Houston
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Word Count: 19654

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5 Linear CHAPTER Algebra 5.1 Introduction Linear algebra is an important mathematical subject independent of its connection to dierential equations. However, it is the relation between dierential equations and linear algebra that is important to us and so we will use this relationship to motivate our study of linear algebra. In Chapter 3 we studied second order linear dierential equations: y + p(x)y + q(x)y = f...

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U. Houston - MATH - 3321
Systems of Linear Dierential EquationsI. Converting a linear equation to a linear systemConsider the third-order equation y or y = r (t)y q (t)y p(t)y + f (t). + p (t )y + q (t )y + r (t )y = f (t )1Introduce new dependent variables x1, x2, x3, as fol
U. Houston - MATH - 3321
CHAPTER 6Linear Dierential Systems6.1 Higher-Order Linear Dierential EquationsThis section is a continuation of Chapter 3. As you will see, all of the theory that we developed for second-order linear dierential equations carries over, essentially verba
U. Houston - MATH - 3321
The Complex Number System The need for extending the real number system is evident when considering solutions of simple equations. For example, the equation x2 + 1 = 0 has no real number solutions, for if x is any real number, then x2 0 and so x2 + 1 1. W
U. Houston - MATH - 3321
MATH 3321 1. ln x dx = x (a) ln(ln x) + C (b) (d)1 2EMCF Quiz 11/20/10(ln x)2 (ln x)2 + C(c) (ln x) + C1 2(e) None of the above.2.dx = 4 + x2 (a) (b) (d)1 2 1 2ln(4 + x2) + C tan1 x + C tan1 (x/2) + C(c) tan1 (x/2) + C1 2(e) None of the abov
U. Houston - MATH - 3321
MATH 3321Quiz 21/26/101. If 2 + 3 = 10, 6 + 5 = 66, 8 + 4 = 96, and 3 + 6 = 27, then 9 + 6 =? (a) 122 (b) 108 (c) 135 (d) 152 (e) None of the above. 2. x dx = 9 x2 x +C 3 (b) (9 x2 )1/2 + C x (c) x sin1 +C 3 (d) 9 x2 + C (a) sin1 (e) None of the above.
U. Houston - MATH - 3321
MATH 3321 1. xy = x2y + 2x3y 2 is (a) a linear equation. (b) a separable equation. (c) a Bernoulli equation. (d) a homogeneous equation. (e) none of the above.Quiz 32/1/102. xy = x3 2x2y is (a) a linear equation. (b) a separable equation. (c) a Bernoul
U. Houston - MATH - 3321
MATH 3321Quiz 42/3/10 1 1 y = 2 + 2, x x y (1) = 0,1. If y = y (x) is the solution of the initial-value problem y + then y (e) =: (a) (b) (c) (d) (e) 1 +e e e 1 e2 1 e e None of the above. dy 3y = x3y 1/3 is: dx2. The general solution of x (a) y 1/3 =
U. Houston - MATH - 3321
MATH 3321Quiz 52/14/101. The general solution of y 2y 15y = 0 is: (a) y = C1 e5x + C2e3x (b) y = C1 e5x + C2 e3x (c) y = C1 e5x + C2e3x (d) y = C1 e5x + C2e3x (e) None of the above.2. The solution of the initial-value problem y + 8y + 16y = 0, y (0) =
U. Houston - MATH - 3321
MATH 3321Quiz 62/22/101. If y = y (x) is a solution of x2y + 6xy + 6y = 0, then lim y (x) = (Hint: the equation x has solutions of the form y = xr .) (a) (b) 1 (c) 0 (d) Does not exist. (e) None of the above. 2. If y = y (x) is the solution of the init
U. Houston - MATH - 3321
MATH 3321Quiz 73/4/101. If f (x) = 2e3x + cos 2x + 3x, then L[f (x)] = (a) (b) (c) (d) (e) 2 3 2 +2 +2 s3 s +4 s s 3 2 +2 +2 s+3 s +4 s s 3 2 +2 +3 s3 s +4 s 2 3 2 +2 + s+3 s +4 s None of the above.2. If f (x) = 2e2x sin 3x 2xe3x + 4, then L[f (x)] =
U. Houston - MATH - 3321
MATH 3321Quiz 810/15/091. Set f (x) = 2xex + 4 cos 2x + 2x. L[f (x)] = 2s 2 2 +2 +3 (s + 1) s + 4 s 4 2 2 +2 (b) +2 2 (s 1) s +4 s 2 2 4s +2 (c) +2 2 (s + 1) s +4 s 4 2 2s +2 (d) +2 (s + 1)3 s + 4 s (a) (e) None of the above.2. Set f (x) = 4 cosh x 3x
U. Houston - MATH - 3321
MATH 3321 1. Set f (x) = x2 + 2, x,Quiz 9 0x&lt;1 x1 1 s2 1 s2 1 s2 1 s2 1 s 2 + s 3 s 2 s + . L[f (x)] =3/22/10(a) (b) (c) (d) (e)3 2 2 + es 3 3 s s s 2 2 2 + es 3 + 3 s s s 2 2 1 + es 3 + 3 s s s 2 2 2 + es 3 3 s s s None of the above.2. Set f (x) =
U. Houston - MATH - 3321
MATH 3321 1 3 1. If A = 2 2 , B = 4 0 then d23 = (a) 1 (b) 17 (c) 9 (d) 17 (e) None of the above. Quiz 10 3 2 2 0 4 24/7/10 2 3 0 , C = 4 2 3 , and D = AB 3C , 0 1 32. If A is 2 3, B is 3 4, C is 4 2, and D is 3 2, then ABC + 2D is: (a) 2 2 (b) 3 2 (c
U. Houston - MATH - 3321
MATH 3321Quiz 114/12/091. If a system of n linear equations in n unknowns is consistent, then the rank of the matrix of coecients is n. (a) Always true (b) Sometimes true (c) False (d) None of the above2. If the determinant of the matrix of coecients
U. Houston - MATH - 3321
MATH 3321Quiz 124/19/101. If a system of n linear equations in n unknowns has innitely many solutions, then the rank of the matrix of coecients is n 1. (a) Always true. (b) Sometimes true. (c) Never true. (d) None of the above.2. If 0 is an eigenvalue
U. Houston - MATH - 3321
MATH 3321Quiz 134/22/101. If 0 is not an eigenvalue of the matrix of coecients of a homogeneous system of n linear equations in n unknowns, then the system has innitely many solutions. (a) Always true. (b) Sometimes true. (c) Never true. (d) None of th
U. Houston - MATH - 3321
Exam 1 Review Show that y = y(x) is/is not a solution of the differential equation F (x, y, y , . . . y (n) = 0. Examples: 1. x2 y - 3x y + 4y = 0; 2. d3y dy + = ex ; 3 dx dx y1 (x) = x2 , y2 (x) = x2 ln xy(x) = 1 + sin x + 1 ex , z(x) = 2 cos x + 1 ex .
U. Houston - MATH - 3321
ANSWERS Exam 1 ReviewPart I. y is/is not a solution. 1. Both are solutions. 2. Both are solutions. 3. y1 is a solution, y2 is not. 4. Both are solutions. 5. y is a solution. Part II. Find the dierential equation. 1. 2xyy = 3y 2 + 6. 2. x2y xy 3y = 0. 3.
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 2 Linear Nonhomogeneous Dierential Equations 1. Find the general solution of y 4 6 4 y + 2 y = 2. x x x2. Find the general solution of y + 4 y = 2 tan 2x. 3. Find the general solution of y 6 y + 9 y = 4 e3x + 4. Find t
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 2Linear Nonhomogeneous Dierential Equations 1. Find the general solution of y Answer y = C1 x2 + C2 x3 + 2 . 3 2. Find the general solution of y + 4 y = 2 tan 2x. Answer y = C1 cos 2x + C2 sin 2x 1 24 6 4 y + 2 y = 2
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 3 2x 4 3 5x 3y 2 2 y 5 2 5 121. Find x and y so that+=.2. Let A =2 1 3 0 4 2,B=3 1 25,3 2 C = 0 1 . 1 2 (b) AB (c) B + AC (d) CBAPerform the indicated operations, if possible: (a) AC3. Show that the matrix
U. Houston - MATH - 3321
MATH 3321 Sample Questions for Exam 31. Find x and y so that2x 4 3 5x+3y 2 2 y=5 2 5 12.Answer: x = 31/17, y = 49/17 2 1 3 0 4 2 3 1 25 3 2 C = 0 1 . 1 2 (b) AB (c) B + AC (d) CBA 2. Let A =,B=,Perform the indicated operations, if possible:
U. Houston - MATH - 3321
MATH 3321 Final Exam Sample Questions 1. y 2 = Cx3 3 is the general solution of a dierential equation. Find the equation. Answer y = 3y 2 + 9 2xy cos 2x . x22. Find the general solution of x y + 3 y = Answer y = sin 2x C + 3. 2x3 x3. Find the general so
U. Houston - MATH - 3321
MATH 3321 ENGINEERING MATHEMATICS Homework Problems Chapter 1 Exercises 1.1: 1, 2, 3, 4, 7, 9, 10, 12, 13, 14, 16, 17, 18, 21, 22 Exercises 1.3: 2, 3, 6, 7, 8, 9, 11, 14, 15, 16, 17, 18, 20, 21, 24 Chapter 2 Exercises 2.1: 2, 3, 4, 7, 9, 11, 14, 17, 18, 2
U. Houston - MATH - 3321
POLYNOMIALSDenition 1: A real polynomial is an expression of the form P (x) = an xn + an1 xn1 + + a1 x + a0 where n is a nonnegative integer and a0 , a1, . . . , an1, an are real numbers with an = 0. The nonnegative integer n is called the degree of P .
U. Houston - MATH - 3321
Higher-Order Linear Differential Equationsnth order linear differential equation: (1) y(n) + pn1( x) y(n1) +L+ p1( x) y '+ p0 ( x) y = f ( x) where p0 , p1, K, pn1, and f are continuous functions on some interval I .(1) is homogeneous if f ( x) 0 on I :
U. Houston - MATH - 3321
Table of Integrals 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. ur du = ur+1 + C, r+1 r = 1 2. 4. 6. 8. 10. 12. 14. 16. 18. 1 u2 + C u2 1| + C 20. 1 du = ln |u| + C u sin u du = cos u + C tan u du = ln | sec u| + C sec u du = ln | sec u + tan u| + C sec u tan u
U. Houston - MATH - 3321
1.If a system of n linear equa-tions in n unknowns is consistent, then the rank of the matrix of coefcients is n.1. Always true2. Sometimes true3. False12.If the determinant of thematrix of coecients of a system of n linear equations in n unknown
U. Houston - POLS - 1336
1 .The US Constitution is the oldest and shortest constitution still in effect today.Student Response Value True 100%Score: 1.5/1.52 .Early Supreme Court decisions (to 1835) expanded the power of the national government at the expense of state govern
U. Houston - POLS - 1336
1 .Match 8th and 9th amendment cases with their proper descriptionStatement Response Value Griswold v. Connecticut 1963 Established Right to Privacy 11.11% Roe v. Wade 1973 Legalized 1st trimester abortions 11.11% Webster 1989 outlawed 1 abortion method
U. Houston - POLS - 1336
1 .While politically significant, talk show format is one of the least popular on radio.Student Response Value False 100%Score: 2/22 .Until the 1980s, there were federal laws whichStud ent Resp onse Valu e A. limit ed how many medi a outlet s one co
U. Houston - POLS - 1336
1 .In Texas, any party which receives at least 20% of the vote for governor must use primary elections to nominate their candidates for public (elected) office.Student Response Value True 100%Score: 1/12 .The Republican front-runner for the 1968 pres
U. Houston - CHEM - 1331
Blackboard Learning System5/9/10 3:39 PMYour location: Home Page Exams Final_Exam Assessments View All Submissions View AttemptView Attempt 1 of 1Title: Started: Submitted: Time spent: Final_Exam May 7, 2010 11:01 AM May 7, 2010 12:55 PM 01:54:11 Tota
U. Houston - CHEM - 1331
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U. Houston - CHEM - 1331
JumptoNavigationFrameViewAttempt1of1Title: Started: Submitted: Timespent: Totalscore: 1.Yourlocation:AssessmentsViewAllSubmissionsViewAttemptMidterm_Exam_1 September28,200911:27AM September28,200912:26PM 00:58:23 18/20=90% Totalscoreadjustedby0.0 Ma
U. Houston - CHEM - 1331
Blackboard Learning System4/4/10 2:04 PMYour location: Home Page Exams Midterm_Exam_2 Assessments View All Submissions View AttemptView Attempt 1 of 1Title: Started: Submitted: Time spent: Midterm_Exam_2 March 30, 2010 1:08 PM March 30, 2010 2:26 PM 0
U. Houston - CHEM - 1331
1. Whichquantitiesarevariablesandwhicharefixedinthefollowing:Charles'slaw StudentResponse A. Vandnarevariables,PandTare fixed B. VandParevariables,nandTare fixed C. PandTarevariables,Vandnare fixed D. VandTarevariables,nandPare fixed E. nandTarevariables
U. Houston - CHEM - 1331
ViewAttempt1of1Title: Started: Submitted: Timespent: Totalscore: MidtermExam2 October28,200911:26AM October28,200912:42PM 01:16:31 19/20=95% Totalscoreadjustedby0.0 Maximumpossiblescore:201. &quot;Thepressureofanidealgasisinverselyproportionaltoitsvolumeatco
U. Houston - CHEM - 1331
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U. Houston - CHEM - 1331
Blackboard Learning System5/1/10 7:02 PMYour location: Home Page Exams Midterm_Exam_3 Assessments View All Submissions View AttemptView Attempt 1 of 1Title: Started: Submitted: Time spent: Midterm_Exam_3 April 28, 2010 1:05 PM April 28, 2010 2:01 PM 0
HCCS - BIOL - 2401
Lab ReportName: _ Section: _Lab Report: Organization of Muscle TissuePurpose: What is the purpose of this exercise? Are there any safety concerns associated with this exercise? If so, list what they are and what precautions should be taken. Activity 1:
HCCS - BIOL - 2401
Lab ReportName: _ Section: _Lab Report: Muscle PhysiologyPurpose: What is the purpose of this exercise? Are there any safety concerns associated with this exercise? If so, list what they are and what precautions should be taken. Refer to Page 15 of thi
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UC Irvine - OCHEM - 51LB
Using Extraction to Separate a MixtureBy: Akash Agarwal, subtitle Sandoval, and Evan Click to edit Master Brittnee style Nahigan2/10/11PurposeOne purpose of this lab was to use the technique of extraction to separate an acid from a mixture of a neutra
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Houston Downtown - CIS - 3301
EasyCommunication+Cheap Technology =AFlattenedWorldAlaa ClicktoeditMastersubtitlestyle CharlaeWilliams AngelLeslie JorgeMartinez HenryRobertson By2/10/11&quot;TheWorldisFlat,&quot;videobyThomasFriedman hadpaintedapictureofthedetailedlandscapeof thecurrentworldth
DeAnza College - PHYS - 10
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