1 Continuation of solutions

Size: px
Start display at page:

Download "1 Continuation of solutions"

Transcription

1 Math 175 Honors ODE I Spring 13 Notes 4 1 Continuation of solutions Theorem 1 in the previous notes states that a solution to y = f (t; y) (1) y () = () exists on a closed interval [ h; h] ; under certain hypotheses on f. However, this is misleading, because usually this interval is not the maximal interval of existence. In most of the cases we will study, f is smooth on all of R ; and in this circumstance, the maximal interval of existence is always an open interval. Furthermore, if the maximal interval of existence is bounded, then the solution becomes unbounded as t approaches either end of this interval. In the following theorem I will only consider solutions for t ; but similar remarks apply for t < : Theorem 1 Suppose in (1) that are continuous in all of R : If a solution of (1)-() exists on [; A] for some A > ; then it can be continued to a solution on some longer interval [; A + h] : Proof. Let R be a rectangle [A a; A + a] [ (A) b; (A) + b] ; where a is small enough that A a > : Consider the initial value problem y = f (t; y) (3) y (A) = (A) : (4) Since f is continuous on the rectangle R; which contains (A; (A)) ; Theorem 1 applied to (3)-(4) with an appropriate shift of the coordinates implies there is a solution of (3)-(4) on some interval [A; A + h] : I claim that the function ^ (t) = (t) if t A (t) if A < t A + h solves ((3)-(4) on [; A + h] : This is obvious if we can show that ^ is di erentiable at A: This is true if the left and right derivatives exist at A and are equal. We can 1

2 compute these. Since (A) = (A), the right derivative is ^ (A + s) ^ (A) lim s! + s (A + s) (A) (A + s) (A) = lim = lim s! + s s! + s = (A) = f (A; (A)) = f (A; (A)) = (A) ; and the left derivative is (A) because is a solution on [; A] : Corollary The maximal interval of existence I is an open interval. Proof. I show that it is open on the right. It is similarly open on the left. If A > is in I, then the theorem shows that some interval (A h; A + h) is also in I; and this is the de nition of an open interval. Theorem 3 If f are continuous R ; and is a solution of (1)-() on some interval I containing ; then is the unique solution on this interval. Proof. Suppose that I = (;!) : Since the initial condition is at t = ; the existence and uniqueness theorem implies that is the unique solution on [ h; h] ; for some h > : We consider uniqueness on [;!); with uniqueness on (; ] proved similarly. Suppose that there are di erent two solutions, 1 and, to (1)-() on [;!): Then 1 = on [; h] ; but there is some T (h;!) with 1 (T ) 6= (T ) : Let = inf ft (;!) j 1 () 6= ()g : Then by its de nition, h < T: In fact, < T because if 1 (T ) 6= (T ) ; then 1 6= on some interval to the left of T; since 1 and are continuous. Also, since 1 and are continuous and 1 = on [; ); we must have 1 () = () : We know, moreover, that the initial value problem y = f (t; y) ; y () = 1 () has a unique solution on some interval [ ; + ] ; with > : But this implies that 1 = on [; + ] ; and this contradicts the de nition of : And one nal result of this type, where we again consider only t : Theorem 4 Under the same hypotheses, if the maximal positive interval of existence of a solution is [;!); where! < 1; then is unbounded in [;!): Proof. Suppose that is bounded on [;!). Then f (t; (t)) is bounded on [;!): Suppose that jf (t; (t))j M on this interval. We have (t) = Z t f (s; (s)) ds:

3 I claim that lim t!! (t) exists. To see this, consider any increasing sequence ft j g converging to!. Then for any j and k > j; Z tk j (t k ) (t j )j = f (s; (s)) ds M jt j t k j : t j Since ft j g is a Cauchy sequence (being convergent), f (t j )g must be a Cauchy sequence, and so also convergent. Since we considered an arbitrary sequence converging to!; lim t!! (t) must exist. We can extend to the point! by letting (!) be the limit. Consider the initial value problem y = f (t; y) y (!) = (!) : This equation has a unique solution on [! h;! + h] for some h > : But this extends the solution a little further, and so [;!] is not the maximal positive interval of existence, a contradiction. This proves the theorem. Remarks on second order linear equations This section summarizes Chapter 3 in the text. in the text. Recall the following de nition: I will not repeat all of the theory De nition 5 Two functions, y 1 (t) and y (t) are called linearly dependent on an interval I if there are two numbers, c 1 and c, such that for all t I, and c 1 + c 6= : c 1 y 1 (t) + c y (t) = (That is a quick way of saying that at least one of the constants is non-zero). Equivalently, y 1 and y are linearly dependent if either y 1 (t) = for all t in I, or there is a constant c (= c 1 c ) such that y (t) = cy 1 (t) for all t in I. Since c might be zero, they are linearly dependent if either function is zero. 3

4 The functions are linearly independent if they are not linearly dependent. In this chapter we consider second order linear equations. First we discuss the homogeneous case, which in greatest generality can be written P (t) y + Q (t) y + R (t) y = : (5) It is assumed that the functions P (t) ; Q (t) ; and R (t) are continuous in an interval I; with P (t) 6= on I. The main theoretical facts are: 1. If y 1 (t) and y (t) are both solutions of (5), and c 1 and c are real numbers, then y (t) = c 1 y 1 (t) + c y (t) is also a solution of (5).. As a consequence, the set V of all possible solutions to (5) is a vector space. It is a subspace of C (I) ; the vector space of all real valued functions which have second derivatives which are continuous on the interval I. 3. The vector space V is of dimension two, or equivalently, if y 1 and y are linearly independent solutions of (5) on I; then every other solution can be written as a linear combination c 1 y + c y. This follows because there is a basis consisting of two linearly independent solutions. As long as the solutions are linearly independent, you can satisfy any set of initial conditions by solving the equations for c 1 and c in terms of and : y () = y () = c 1 y 1 () + c y () = c 1 y 1 () + c y () = 4. Operator notation is introduced. If is a function such that and exist in I, let L denote the new function de ned by (L) (t) = P (t) (t) + Q (t) (t) + R (t) (t) : Then the ode (5) can be written as Ly = : 4

5 L is called a di erential operator. It is often a convenient shorthand in studying (5). Since we assumed that P (t) is not zero in I; we can divide the equation by P (t) and get one of the form y + p (t) y + q (t) y = : This gives a di erent operator L; used for example on page 143 (145 in the 9th ed.). The text uses slightly di erent notation from above, writing L [] = + p + q: 5. A solution of Ly = is completely determined by the values of y (t ) and y (t ) at a point in I. Thus, we study initial value problems of the following kind: y + p (t) y + q (t) y = y (t ) = y (t ) = ; where and are given real numbers. Item (3) above is equivalent to an existence and uniqueness theorem for this initial value problem. We will discuss the proof when we get to Chapter 7. (On page 148 (15), equation (13), the initial conditions are written as y (t ) = y ; y (t ) = y. Thus, in particular, y denotes a real number, which I called above.) 6. There is a test for linear independence called the Wronskian test. If y 1 (t) and y (t) are solutions of (5), then the Wronskian of these two solutions is the function W (t) = y 1 (t) y (t) y (t) y1 (t) : The function W and consequently, satis es the rst order linear equation W = p (t) W; W (t) = ce R p(t)dt : (Abel s formula) The solutions y 1 and y are linearly independent on I if and only if c 6= : Here c can be found using the initial conditions on y 1 and y.) 5

6 Most of the rest of this chapter is devoted to nding exact solutions to the case where P; Q; and R are constants. We can write this as ay + by + cy = : The basic method is to substitute y = e rt into the ode, obtaining with solutions ar + br + c = ; r = b p b 4ac : a There are several cases to consider..1 real unequal roots case This is when b 4ac > : Two solutions are then y 1 = e r 1t y = e r t : We can easily show that these are linearly independent, since y 1 y and this is not constant. = er 1 t e r t = e (r 1 r )t ;. real equal roots case This is when b 4ac = ; and r 1 = r : We saw in class that the two linearly independent solutions are y 1 = e r 1t y = te r 1t : 6

7 .3 complex roots case Here we assume that the roots are r = + i: Here, = b p 4ac b a ; = : a I claimed that the two linearly independent solutions are y 1 = e t cos t y = e t sin t: This was obtained through some mysterious calculations with complex numbers, so we need to check that it is right, which I now will do, for y 1 : so But so Also, y = e t cos t y = e t cos t e t sin t y = e t cos t e t sin t ay + by + cy = e t a + b + c cos t + ( a b) sin t a = b (4ac b ) 4a 4a b (4ac b ) b + b + c = a 4a 4a a + c a b = a = b 4a = p 4ac b b a c + b 4a p b 4ac b a a b a b a + c = : b = : p 4ac b This proves that the solution works, irrespective of how it was obtained. a 7

8 3 Qualitative behavior of solutions in the di erent cases. 3.1 real unequal roots. Suppose, for example, that y 1 = e t y = e t : Then y 1 is monotone increasing, while y is monotone decreasing. Linear combinations may not be monotone, however. For example, 1 et + 1 e t tends to 1 as t! 1 and also as t! 1. This function is common enough to have a name, cosh t, with graph y x 3. equal roots Here we saw that one example has solutions y 1 = e t y = te t : Here are their graphs: y x

9 (You should be able to tell which is which). 3.3 complex roots Since these solutions involve sines and cosines, the solutions oscillate. is e t 1 cos t One sample y x 1 4 Homework Due at the beginning of class on Jan Give a careful explanation of the second sentence in the proof of theorem 4.. (a) Find a function which is di erentiable and unbounded on [; 1) but for which there is an increasing sequence ft j g [; 1) with (t j ) = : (b) Can such a function be the solution of an equation y = f (t; y) on [; 1) if f is continuous on R? Justify your answer. 3. (worth 15 points) (a) Use De nitions 1 and from the Notes on Uniform Convergence and equation () in nots # 3 to prove that uniformly in the interval t h. lim j j+1 (t) j (t)j = j!1 (b) Prove that lim n!1 f (s; n (s)) = f (s; (s)) uniformly for s h: Hint: Use the Lipschitz constant L and what was already proved about the sequence f n (s)g : Hint: These are "" K (") proofs, in the terminology of Bartel and Sherbert. That is, to apply De nition you need to show that for each " > there is an 9

10 appropriate K (") as used in the de nition of convergence of a sequence, where you consider n K ("). You may use the fact that the Taylor series for e x converges for all x. 4. Derive the general solution of y 4y + 4y = t + e t : Do two problems and add the particular solutions. You may have to read the book, or use the table in section 3.6 (3.5). (I won t grade whether you check your nal answer but you should!) 1

Exam II Review: Selected Solutions and Answers

Exam II Review: Selected Solutions and Answers November 9, 2011 Exam II Review: Selected Solutions and Answers NOTE: For additional worked problems see last year s review sheet and answers, the notes from class, and your text. Answers to problems from

More information

1 Some general theory for 2nd order linear nonhomogeneous

1 Some general theory for 2nd order linear nonhomogeneous Math 175 Honors ODE I Spring, 013 Notes 5 1 Some general theory for nd order linear nonhomogeneous equations 1.1 General form of the solution Suppose that p; q; and g are continuous on an interval I; and

More information

1 Which sets have volume 0?

1 Which sets have volume 0? Math 540 Spring 0 Notes #0 More on integration Which sets have volume 0? The theorem at the end of the last section makes this an important question. (Measure theory would supersede it, however.) Theorem

More information

Second Order Linear Equations

Second Order Linear Equations Second Order Linear Equations Linear Equations The most general linear ordinary differential equation of order two has the form, a t y t b t y t c t y t f t. 1 We call this a linear equation because the

More information

Math 1270 Honors Fall, 2008 Background Material on Uniform Convergence

Math 1270 Honors Fall, 2008 Background Material on Uniform Convergence Math 27 Honors Fall, 28 Background Material on Uniform Convergence Uniform convergence is discussed in Bartle and Sherbert s book Introduction to Real Analysis, which was the tet last year for 42 and 45.

More information

1 A complete Fourier series solution

1 A complete Fourier series solution Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider

More information

Second-Order Linear ODEs

Second-Order Linear ODEs Second-Order Linear ODEs A second order ODE is called linear if it can be written as y + p(t)y + q(t)y = r(t). (0.1) It is called homogeneous if r(t) = 0, and nonhomogeneous otherwise. We shall assume

More information

Second Order Differential Equations Lecture 6

Second Order Differential Equations Lecture 6 Second Order Differential Equations Lecture 6 Dibyajyoti Deb 6.1. Outline of Lecture Repeated Roots; Reduction of Order Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters

More information

4.3 - Linear Combinations and Independence of Vectors

4.3 - Linear Combinations and Independence of Vectors - Linear Combinations and Independence of Vectors De nitions, Theorems, and Examples De nition 1 A vector v in a vector space V is called a linear combination of the vectors u 1, u,,u k in V if v can be

More information

6.2 Important Theorems

6.2 Important Theorems 6.2. IMPORTANT THEOREMS 223 6.2 Important Theorems 6.2.1 Local Extrema and Fermat s Theorem Definition 6.2.1 (local extrema) Let f : I R with c I. 1. f has a local maximum at c if there is a neighborhood

More information

Math 322. Spring 2015 Review Problems for Midterm 2

Math 322. Spring 2015 Review Problems for Midterm 2 Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly

More information

1 The relation between a second order linear ode and a system of two rst order linear odes

1 The relation between a second order linear ode and a system of two rst order linear odes Math 1280 Spring, 2010 1 The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00

More information

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm.

Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. 1 competing species Math 1280 Notes 4 Last section revised, 1/31, 9:30 pm. This section and the next deal with the subject of population biology. You will already have seen examples of this. Most calculus

More information

Chapter 4: Higher Order Linear Equations

Chapter 4: Higher Order Linear Equations Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order

More information

MATH 1190 Exam 4 (Version 2) Solutions December 1, 2006 S. F. Ellermeyer Name

MATH 1190 Exam 4 (Version 2) Solutions December 1, 2006 S. F. Ellermeyer Name MATH 90 Exam 4 (Version ) Solutions December, 006 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.

More information

(x + y) ds. 2 (1) dt = p Find the work done by the force eld. yzk

(x + y) ds. 2 (1) dt = p Find the work done by the force eld. yzk MATH Final Exam (Version 1) Solutions May 4, 11 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.

More information

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا

More information

Answer Key b c d e. 14. b c d e. 15. a b c e. 16. a b c e. 17. a b c d. 18. a b c e. 19. a b d e. 20. a b c e. 21. a c d e. 22.

Answer Key b c d e. 14. b c d e. 15. a b c e. 16. a b c e. 17. a b c d. 18. a b c e. 19. a b d e. 20. a b c e. 21. a c d e. 22. Math 20580 Answer Key 1 Your Name: Final Exam May 8, 2007 Instructor s name: Record your answers to the multiple choice problems by placing an through one letter for each problem on this answer sheet.

More information

5.5 Deeper Properties of Continuous Functions

5.5 Deeper Properties of Continuous Functions 5.5. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 195 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested

More information

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS)

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS) Name: November 8, 011 Worksheet # : Higher Order Linear ODEs (SOLUTIONS) 1. A set of n-functions f 1, f,..., f n are linearly independent on an interval I if the only way that c 1 f 1 (t) + c f (t) +...

More information

Second Order Linear Equations

Second Order Linear Equations October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d

More information

Work sheet / Things to know. Chapter 3

Work sheet / Things to know. Chapter 3 MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients

More information

Homework #6 Solutions

Homework #6 Solutions Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly

More information

Work sheet / Things to know. Chapter 3

Work sheet / Things to know. Chapter 3 MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients

More information

Lecture Notes for Math 251: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation

Lecture Notes for Math 251: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation Lecture Notes for Math 21: ODE and PDE. Lecture 12: 3.3 Complex Roots of the Characteristic Equation Shawn D. Ryan Spring 2012 1 Complex Roots of the Characteristic Equation Last Time: We considered the

More information

Monday, 6 th October 2008

Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns Monday, 6 th October 2008 MA211 Lecture 9: 2nd order differential eqns 1/19 Class test next week... MA211 Lecture 9: 2nd order differential eqns 2/19 This morning

More information

Section 3.1 Second Order Linear Homogeneous DEs with Constant Coefficients

Section 3.1 Second Order Linear Homogeneous DEs with Constant Coefficients Section 3. Second Order Linear Homogeneous DEs with Constant Coefficients Key Terms/ Ideas: Initial Value Problems Homogeneous DEs with Constant Coefficients Characteristic equation Linear DEs of second

More information

MAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES.

MAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES. MAT9 - Calculus III - Fall 4 Solution for Term Test - November 6, 4 Time allotted: 9 minutes. Aids permitted: None. Full Name: Last First Student ID: Email: @mail.utoronto.ca Instructions DO NOT WRITE

More information

MATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name

MATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name MATH 223 Final Exam Solutions ecember 14, 25 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In

More information

APPM 2360: Midterm 3 July 12, 2013.

APPM 2360: Midterm 3 July 12, 2013. APPM 2360: Midterm 3 July 12, 2013. ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor s name, (3) your recitation section number and (4) a grading table. Text books, class notes,

More information

6.2 Deeper Properties of Continuous Functions

6.2 Deeper Properties of Continuous Functions 6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 69 6.2 Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in

More information

Linear Independence and the Wronskian

Linear Independence and the Wronskian Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions

More information

8 Periodic Linear Di erential Equations - Floquet Theory

8 Periodic Linear Di erential Equations - Floquet Theory 8 Periodic Linear Di erential Equations - Floquet Theory The general theory of time varying linear di erential equations _x(t) = A(t)x(t) is still amazingly incomplete. Only for certain classes of functions

More information

CHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s

CHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY DIFFERENTIAL

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Additional Homework Problems

Additional Homework Problems Additional Homework Problems These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a differential

More information

µ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition

µ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following

More information

Homework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator.

Homework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator. Homework 3 1 If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator Solution: Assuming that the inverse of T were defined, then we will have to have that D(T 1

More information

Math 216 Second Midterm 16 November, 2017

Math 216 Second Midterm 16 November, 2017 Math 216 Second Midterm 16 November, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material

More information

1 More concise proof of part (a) of the monotone convergence theorem.

1 More concise proof of part (a) of the monotone convergence theorem. Math 0450 Honors intro to analysis Spring, 009 More concise proof of part (a) of the monotone convergence theorem. Theorem If (x n ) is a monotone and bounded sequence, then lim (x n ) exists. Proof. (a)

More information

Homogeneous Linear Systems and Their General Solutions

Homogeneous Linear Systems and Their General Solutions 37 Homogeneous Linear Systems and Their General Solutions We are now going to restrict our attention further to the standard first-order systems of differential equations that are linear, with particular

More information

Homogeneous Equations with Constant Coefficients

Homogeneous Equations with Constant Coefficients Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form

More information

Nonhomogeneous Equations and Variation of Parameters

Nonhomogeneous Equations and Variation of Parameters Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential

More information

Math 163 (23) - Midterm Test 1

Math 163 (23) - Midterm Test 1 Name: Id #: Math 63 (23) - Midterm Test Spring Quarter 208 Friday April 20, 09:30am - 0:20am Instructions: Prob. Points Score possible 26 2 4 3 0 TOTAL 50 Read each problem carefully. Write legibly. Show

More information

6. Linear Differential Equations of the Second Order

6. Linear Differential Equations of the Second Order September 26, 2012 6-1 6. Linear Differential Equations of the Second Order A differential equation of the form L(y) = g is called linear if L is a linear operator and g = g(t) is continuous. The most

More information

Math 273 (51) - Final

Math 273 (51) - Final Name: Id #: Math 273 (5) - Final Autumn Quarter 26 Thursday, December 8, 26-6: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all

More information

Midterm 1. Every element of the set of functions is continuous

Midterm 1. Every element of the set of functions is continuous Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions

More information

MATH 308 Differential Equations

MATH 308 Differential Equations MATH 308 Differential Equations Summer, 2014, SET 6 JoungDong Kim Set 6: Section 3.3, 3.4, 3.5, 3.6 Section 3.3 Complex Roots of the Characteristic Equation Recall that a second order ODE with constant

More information

Linear Second Order ODEs

Linear Second Order ODEs Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that

More information

Stochastic Processes

Stochastic Processes Stochastic Processes A very simple introduction Péter Medvegyev 2009, January Medvegyev (CEU) Stochastic Processes 2009, January 1 / 54 Summary from measure theory De nition (X, A) is a measurable space

More information

Chapter 13: General Solutions to Homogeneous Linear Differential Equations

Chapter 13: General Solutions to Homogeneous Linear Differential Equations Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)

More information

Math 333 Qualitative Results: Forced Harmonic Oscillators

Math 333 Qualitative Results: Forced Harmonic Oscillators Math 333 Qualitative Results: Forced Harmonic Oscillators Forced Harmonic Oscillators. Recall our derivation of the second-order linear homogeneous differential equation with constant coefficients: my

More information

APPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014

APPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014 APPM 2360 Section Exam 3 Wednesday November 9, 7:00pm 8:30pm, 204 ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) lecture section, (4) your instructor s name, and (5)

More information

5.5 Deeper Properties of Continuous Functions

5.5 Deeper Properties of Continuous Functions 200 CHAPTER 5. LIMIT AND CONTINUITY OF A FUNCTION 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested

More information

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

More information

Math 4263 Homework Set 1

Math 4263 Homework Set 1 Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that

More information

Linear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order

Linear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order Linear Homogeneous ODEs of the Second Order with Constant Coefficients. Reduction of Order October 2 6, 2017 Second Order ODEs (cont.) Consider where a, b, and c are real numbers ay +by +cy = 0, (1) Let

More information

FENGBO HANG AND PAUL C. YANG

FENGBO HANG AND PAUL C. YANG Q CURVATURE ON A CLASS OF 3 ANIFOLDS FENGBO HANG AND PAUL C. YANG Abstract. otivated by the strong maximum principle for Paneitz operator in dimension 5 or higher found in [G] and the calculation of the

More information

A lower bound for X is an element z F such that

A lower bound for X is an element z F such that Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F

More information

Rings, Integral Domains, and Fields

Rings, Integral Domains, and Fields Rings, Integral Domains, and Fields S. F. Ellermeyer September 26, 2006 Suppose that A is a set of objects endowed with two binary operations called addition (and denoted by + ) and multiplication (denoted

More information

Section 4.7: Variable-Coefficient Equations

Section 4.7: Variable-Coefficient Equations Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find

More information

3.5 Undetermined Coefficients

3.5 Undetermined Coefficients 3.5. UNDETERMINED COEFFICIENTS 153 11. t 2 y + ty + 4y = 0, y(1) = 3, y (1) = 4 12. t 2 y 4ty + 6y = 0, y(0) = 1, y (0) = 1 3.5 Undetermined Coefficients In this section and the next we consider the nonhomogeneous

More information

The Laplace Transform

The Laplace Transform C H A P T E R 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. For such problems the methods described

More information

Transition Density Function and Partial Di erential Equations

Transition Density Function and Partial Di erential Equations Transition Density Function and Partial Di erential Equations In this lecture Generalised Functions - Dirac delta and heaviside Transition Density Function - Forward and Backward Kolmogorov Equation Similarity

More information

Math K (24564) - Lectures 02

Math K (24564) - Lectures 02 Math 39100 K (24564) - Lectures 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Second Order Linear Equations, B & D Chapter 4 Second Order Linear Homogeneous

More information

Fixed Point Theorem and Sequences in One or Two Dimensions

Fixed Point Theorem and Sequences in One or Two Dimensions Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask

More information

MA261-A Calculus III 2006 Fall Homework 7 Solutions Due 10/20/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 7 Solutions Due 10/20/2006 8:00AM MA26-A Calculus III 2006 Fall Homework 7 Solutions Due 0/20/2006 8:00AM 3 #4 Find the rst partial derivatives of the function f (; ) 5 + 3 3 2 + 3 4 f (; ) 5 4 + 9 2 2 + 3 4 f (; ) 6 3 + 2 3 3 #6 Find

More information

MATH 308 Differential Equations

MATH 308 Differential Equations MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this

More information

2. Higher-order Linear ODE s

2. Higher-order Linear ODE s 2. Higher-order Linear ODE s 2A. Second-order Linear ODE s: General Properties 2A-1. On the right below is an abbreviated form of the ODE on the left: (*) y + p()y + q()y = r() Ly = r() ; where L is the

More information

Short Solutions to Review Material for Test #2 MATH 3200

Short Solutions to Review Material for Test #2 MATH 3200 Short Solutions to Review Material for Test # MATH 300 Kawai # Newtonian mechanics. Air resistance. a A projectile is launched vertically. Its height is y t, and y 0 = 0 and v 0 = v 0 > 0. The acceleration

More information

HW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]

HW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,

More information

MATH 425, HOMEWORK 5, SOLUTIONS

MATH 425, HOMEWORK 5, SOLUTIONS MATH 425, HOMEWORK 5, SOLUTIONS Exercise (Uniqueness for the heat equation on R) Suppose that the functions u, u 2 : R x R t R solve: t u k 2 xu = 0, x R, t > 0 u (x, 0) = φ(x), x R and t u 2 k 2 xu 2

More information

MATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam

MATH 251 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam MATH 51 Week 6 Not collected, however you are encouraged to approach all problems to prepare for exam A collection of previous exams could be found at the coordinator s web: http://www.math.psu.edu/tseng/class/m51samples.html

More information

Assignment 3. Section 10.3: 6, 7ab, 8, 9, : 2, 3

Assignment 3. Section 10.3: 6, 7ab, 8, 9, : 2, 3 Andrew van Herick Math 710 Dr. Alex Schuster Sept. 21, 2005 Assignment 3 Section 10.3: 6, 7ab, 8, 9, 10 10.4: 2, 3 10.3.6. Prove (3) : Let E X: Then x =2 E if and only if B r (x) \ E c 6= ; for all all

More information

Math 341 Fall 2008 Friday December 12

Math 341 Fall 2008 Friday December 12 FINAL EXAM: Differential Equations Math 341 Fall 2008 Friday December 12 c 2008 Ron Buckmire 1:00pm-4:00pm Name: Directions: Read all problems first before answering any of them. There are 17 pages in

More information

GALOIS THEORY I (Supplement to Chapter 4)

GALOIS THEORY I (Supplement to Chapter 4) GALOIS THEORY I (Supplement to Chapter 4) 1 Automorphisms of Fields Lemma 1 Let F be a eld. The set of automorphisms of F; Aut (F ) ; forms a group (under composition of functions). De nition 2 Let F be

More information

Lecture 11: Eigenvalues and Eigenvectors

Lecture 11: Eigenvalues and Eigenvectors Lecture : Eigenvalues and Eigenvectors De nition.. Let A be a square matrix (or linear transformation). A number λ is called an eigenvalue of A if there exists a non-zero vector u such that A u λ u. ()

More information

Topic 5 Notes Jeremy Orloff. 5 Homogeneous, linear, constant coefficient differential equations

Topic 5 Notes Jeremy Orloff. 5 Homogeneous, linear, constant coefficient differential equations Topic 5 Notes Jeremy Orloff 5 Homogeneous, linear, constant coefficient differential equations 5.1 Goals 1. Be able to solve homogeneous constant coefficient linear differential equations using the method

More information

23 Elements of analytic ODE theory. Bessel s functions

23 Elements of analytic ODE theory. Bessel s functions 23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2

More information

Math 23: Differential Equations (Winter 2017) Midterm Exam Solutions

Math 23: Differential Equations (Winter 2017) Midterm Exam Solutions Math 3: Differential Equations (Winter 017) Midterm Exam Solutions 1. [0 points] or FALSE? You do not need to justify your answer. (a) [3 points] Critical points or equilibrium points for a first order

More information

Midterm 1 Solutions Math Section 55 - Spring 2018 Instructor: Daren Cheng

Midterm 1 Solutions Math Section 55 - Spring 2018 Instructor: Daren Cheng Midterm 1 Solutions Math 20250 Section 55 - Spring 2018 Instructor: Daren Cheng #1 Do the following problems using row reduction. (a) (6 pts) Let A = 2 1 2 6 1 3 8 17 3 5 4 5 Find bases for N A and R A,

More information

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is

More information

Midterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10

Midterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10 Midterm 1 NAME: RULES: You will be given the entire period (1PM-3:10PM) to complete the test. You can use one 3x5 notecard for formulas. There are no calculators nor those fancy cellular phones nor groupwork

More information

Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 4 Solutions Please write neatly, and in complete sentences when possible.

Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 4 Solutions Please write neatly, and in complete sentences when possible. Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 4 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 2.6.3, 2.7.4, 2.7.5, 2.7.2,

More information

LINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x)

LINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x) LINEAR DIFFERENTIAL EQUATIONS MINSEON SHIN 1. Existence and Uniqueness Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose p 1 (x),..., p n (x) and g(x) are continuous real-valued

More information

First and Second Order Differential Equations Lecture 4

First and Second Order Differential Equations Lecture 4 First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence

More information

1. (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. Solution: Such a graph is shown below.

1. (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. Solution: Such a graph is shown below. MATH 9 Eam (Version ) Solutions November 7, S. F. Ellermeer Name Instructions. Your work on this eam will be graded according to two criteria: mathematical correctness and clarit of presentation. In other

More information

Sign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.

Sign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2. Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave

More information

Higher Order Linear Equations Lecture 7

Higher Order Linear Equations Lecture 7 Higher Order Linear Equations Lecture 7 Dibyajyoti Deb 7.1. Outline of Lecture General Theory of nth Order Linear Equations. Homogeneous Equations with Constant Coefficients. 7.2. General Theory of nth

More information

A: Brief Review of Ordinary Differential Equations

A: Brief Review of Ordinary Differential Equations A: Brief Review of Ordinary Differential Equations Because of Principle # 1 mentioned in the Opening Remarks section, you should review your notes from your ordinary differential equations (odes) course

More information

Linear algebra and differential equations (Math 54): Lecture 19

Linear algebra and differential equations (Math 54): Lecture 19 Linear algebra and differential equations (Math 54): Lecture 19 Vivek Shende April 5, 2016 Hello and welcome to class! Previously We have discussed linear algebra. This time We start studying differential

More information

Math 104: Homework 7 solutions

Math 104: Homework 7 solutions Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for

More information

Math 361: Homework 1 Solutions

Math 361: Homework 1 Solutions January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

More information

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential

More information

Higher Order Linear Equations

Higher Order Linear Equations C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid

More information

Lecture Notes in Mathematics. Arkansas Tech University Department of Mathematics

Lecture Notes in Mathematics. Arkansas Tech University Department of Mathematics Lecture Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Ordinary Differential Equations for Physical Sciences and Engineering Marcel B. Finan c All Rights

More information

Problem List MATH 5143 Fall, 2013

Problem List MATH 5143 Fall, 2013 Problem List MATH 5143 Fall, 2013 On any problem you may use the result of any previous problem (even if you were not able to do it) and any information given in class up to the moment the problem was

More information

Nonlinear Programming (NLP)

Nonlinear Programming (NLP) Natalia Lazzati Mathematics for Economics (Part I) Note 6: Nonlinear Programming - Unconstrained Optimization Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and Blume

More information

Robust Estimation and Inference for Extremal Dependence in Time Series. Appendix C: Omitted Proofs and Supporting Lemmata

Robust Estimation and Inference for Extremal Dependence in Time Series. Appendix C: Omitted Proofs and Supporting Lemmata Robust Estimation and Inference for Extremal Dependence in Time Series Appendix C: Omitted Proofs and Supporting Lemmata Jonathan B. Hill Dept. of Economics University of North Carolina - Chapel Hill January

More information

3.4 Using the First Derivative to Test Critical Numbers (4.3)

3.4 Using the First Derivative to Test Critical Numbers (4.3) 118 CHAPTER 3. APPLICATIONS OF THE DERIVATIVE 3.4 Using the First Derivative to Test Critical Numbers (4.3) 3.4.1 Theory: The rst derivative is a very important tool when studying a function. It is important

More information