First Midterm Exam Name: Practice Problems September 19, x = ax + sin x.

Size: px
Start display at page:

Download "First Midterm Exam Name: Practice Problems September 19, x = ax + sin x."

Transcription

1 Math 54 Treibergs First Midterm Exam Name: Practice Problems September 9, 24 Consider the family of differential equations for the parameter a: (a Sketch the phase line when a x ax + sin x (b Use the graphs of ax and sin x to determine the qualitative behavior of all bifurcations that occur as a increases form to (c Sketch the bifurcation diagram for this family of differential equations The equations y sin x and y ax for a ±, ±3, ±5 are superimposed The zeros of ax + sin x are the intersection points So when a the rest points are at πk for integer k and the flow directions alternate in each interval As a moves from zero, the line y ax intersects y sin x at finitely many and fewer and fewer points When a then there are only five rest points The stable/unstable pairs move toward each other as a increases and vanish The bifurcation diagram are the solutions of a + sin x x, which are plotted as the blue and red curves It shows how as a departs from a and moves to a, there are fewer and fewer rest points that such that sources and sinks cancel pairwise as a increases For a > near a there are only three rest points which collapse to one in a pitchfork bifurcation at x and a After a there is only one rest point at

2 2 Solve the initial value problem: X 5 3 X; X( det(a λi ( 5 λ( λ 3 9 λ 2 + 4λ 32 (λ 4(λ + 8 Hence the eigenvalues are λ 4 and λ 2 8 The eigenvactors is 9 3 (A λ IV, (A λ 2 IV Then the general solution is X(t c e 4t ( 3 + c 2 e 8t ( Then the initial value problem is solved by ( 5 X( 6 3 ( c c 2 c 4, c Find the general solution: X 4 X 2 2 det(a λi ( 4 λ( λ ( 2 λ 2 + 6λ + Hence the eigenvalues are λ 3 ± i An eigenvactor for λ 3 + i is i (A λiv 2 i + i A complex solution is X(t e ( 3+it + i e 3t ( cos t cos t sin t ( e 3t (cos t + i sin t + i + ie 3t ( sin t cos t + sin t The real and imaginary parts are independent solutions so the general solution is cos t sin t X(t c e 3t + c 2 e 3t cos t sin t cos t + sin t 4 Find a matrix T that brings the equation into canonical form Show that your change of variables does the job 4 4 X X det(a λi (4 λ( λ ( 4 λ 2 4λ + 4 (λ 2 2 2

3 Hence the eigenvalues are λ 2, 2 An eigenvactor for λ 2 is (A λiv 2 Since the matrix A λi has rank one, there are no more independent eigenvectors Then the recipe says take any other independent vector, say W (, and compute 2 AW ( µv + νw 2 Then consider the matrix T whose columns are V and (/µw Checking, ( T AT 2 ( To see if it does the job, change variables by X T Y Then 4 4 T Y X X AX AT Y Y T AT Y Y 2 5 Show that the two systems are topologically conjugate by finding a conjugating homeomorphism and checking: X X; Y Y 6 5 Computing the eigenvalues of the first system we find the characteristic equation is det(a λi (8 λ( λ 3 ( 6 λ 2 7λ + (λ 2(λ 5, thus the eigenvalues are λ 2 and λ 2 5 Thus the problem asks that we show that the canonical form is topologically conjugate to the original matrix This is achieved by a linear map that changes variables Computing the eigenvectors we find (A λ IV ( , (A λ 2 IV 2 ( The general solution is thus X(t c e ( + c 2 e 5t The particular solution when X( ( α β is gotten by solving α c c α β, c 2 2α + β β c 2 Then the first flow may be written ( 2e ϕ A t (α, β (α + βe + (2α + βe 5t 5t e e 5t e α 2e 2e 5t 2e e 5t β The second equation is already diagonal, so the flow may be written γe ϕ B e γ t (γ, δ δe 5t e 5t δ 3

4 Putting T ( we have T AT ( 2 ( Letting h(x T X we must show h ϕ A t (x ϕ B t h(x where ϕ A t whereas is the flow of the first system Thus ( 2e h ϕ A 5t e e 5t e α t (α, β 2 2e 2e 5t 2e e 5t β ( e e α 2e 5t e 5t β (α + βe (2α + βe 5t e ϕ B α t h(x e 5t 2 β e α β e 5t 2α + β (α + βe (2α + βe 5t Both flows are the same! 6 Consider the system X a b X b a Sketch the region in the a-b plane where this system has different types of cononical forms Find these canonical forms Show the corresponding regions on the determinant-trace plane det(a λi (a λ 2 b 2 a λ ±b so that eigenvalues are a ± b If b > a then the eigenvalues are negative and positive, and the flow is a saddle with canonical form ( a+b a b If a > b then both eigenvalues are positive, and the flow is an unstable improper node If a < b then both roots are negative and the flow is a stable improper node Along a b > roots are zero and positive, thus flow is an unstable brush with canonical form ( a Along a b < the roots are zero and negative, thus flow is a stable brush If b then the node is a proper node with canonical form ( a a At a b all points are rest points with canonical form If det a 2 b 2 < then roots are opposite signe and the solution is a saddle If det > then the solution is a node, unstable if tr > and stable if tr < If det then one root is zero and the other has the sign of a 4

5 7 Sketch the phase line and the bifurcation diagram corresponding to the family of differential equations with parameter a Find all equilibrium solutions and determine whether they are sinks, sources or neither x x 2 + ax + The roots are 2 a ± 2 a2 4 Thus bifurcation occurs at a ±2 Stable and unstable nodes split as a > 2 increases When a > 2 the left rest point is stable and the right is unstable For a < 2 there are no rest points here is a typical phase line: Plotting the bifurcation diagram we get solving for a ( + x 2 /x The curves locate the rest points at given a 5

6 8 Consider the harmonic oscillator equation with parameters c and k > x + cx + kx (a For which values of c and k does the system have complex eigenvalues? real and distinct eigenvalues? Repeated eigenvalues? identify the regions in the ck-plane where the system has similar phase phase portraits (b In each of the cases in (a, sketch the graph showing the motion of the mass when the mass is released from an initial position with x and zero velocity and from an initial position with x and unit velocity Put x y to get system x x y k c y det(a λi λ( c λ + k λ 2 + cλ + k λ c ± c 2 4k 2 so that the roots are complex conjugate, repeated or real and distinct depending on whether c 2 4k is negative, zero, or positive, resp In the ck-plane If the roots are complex, the spring system is underdamped and the solution from either condition oscillates infinitely often If the roots are repeated the system is critically damped If the roots are real distinct, they are both negative and the system is overdamped In the critically and overdamped cases, the solution may overshoot x at most once However, with the given initial conditions, in these cases the solution returns to x monotonically eg, for an overdamped example c 5 and k 4 then λ 4, so the general solution is X(t c e t ( + c 2 e 4t ( 4 Thus with initial conditions X( ( ( or the solutions are X(t 4 3 e t ( 3 e 4t ( 4 ; X(t 3 e t ( 3 e 4t ( 4 6

7 For the critically damped example c 4 and k 4 then λ, so the general solution is t X(t c e t + c 2 e t Thus with initial conditions X( ( ( or the solutions are t t X(t e t + 2e t ; X(t e t For the overdamped example c 2 and k 4 then λ ± 3i so the general solution is ( X(t c e t cos( ( 3t cos( 3t 3 sin( + c 2 e t sin( 3t 3t 3 cos( 3t sin( 3t Thus with initial conditions X( ( or ( the solutions are ( cos( 3t + 3 sin( 3t X(t e t ( sin( ; X(t ( e t sin( 3t 3t 3 3 cos( 3t sin( 3t The graphs of the solutions beginning from x and x for over/critically/under damped are green/blue/red as spring constant is constant k 4 and the drag is reduced c 5 to 4 to 2 For the same constants, the graphs of solutions beginning with x and x over/critically/under damped are different colors red/blue/green for 9 Find the general solution Show that there is a unique periodic solution Find the Poincaré Map : {t } {t 2π} and use it to verify again that there is a unique 2π - periodic solution x x cos t 7

8 Using integrating factors we find ( e t x e t (x x e t cos t so x(t e t x + t e t s cos s ds e t x + 2 (et cos t + sin t In order that this be periodic, we need exponential terms to cancel, which happens if x 2 To see this solution is unique, since et is nonzero we may suppose that it has the form y(t e t f(t + 2 (et cos t + sin t Since it satisfies the ODE we compute y y cos t e t f (t from which follows f so f(t c, a constant Thus all solutions have this form The Poincaré map (x is the value of the solution with x( x at time t 2π Thus (x e 2π x + 2( e 2π cos(2π + sin(2π e 2π x + 2 (e2π A solution is 2π periodic iff x (x Solving this equation we find the only solution is x 2 which corresponds to the only periodic solution Find the general solution of X AX Assume there is a matrix T such that 3 T 3 ; T AT 2 2 We see that the matrix T diagonalizes the system The columns of T are eigenvectors with corresponding diagonal elements of T AT the eigenvalues, so that the general solution is X(t c e 3t 3 + c 2 e 2 Let A be a 2 2 real matrix Suppose f(x Ax : R 2 R 2 is onto Then A is invertible To show that A is invertible, we shall construct a matrix such that AB I Since f(x Ax is onto, there are vectors V and V 2 such that AV and AV2 ( Let B be the matrix whose columns are V and V 2 Thus AB A(V ; V 2 (AV ; AV 2 It follows that B A To see this, we have det(ab det(a det(b det(i so that det(a and so A can be given by the formula A d b a b where A det(a c a c d By premultiplying AB I by A we find A AB IB B A 8

9 2 Let A and B be 2 2 real matrices Show det(ab det(a det(b This one is just a computation for the 2 2 case Letting a b e f A and B c d g h we have so On the other hand Thus det(a ad bc; det(b eh fg det(a det(b (ad bc(eh fg adeh adfg bceh + bcfg ae + bg af + bh AB ce + dg cf + dh det(ab (ae + bg(cf + dh (af + bh(ce + dg so is equal to det(a det(b acef + adeh + cbfg + bdgh acef adfg bceh bdgh adeh + bcfg adfg bceh 9

Linear Algebra Miscellaneous Proofs to Know

Linear Algebra Miscellaneous Proofs to Know Linear Algebra Miscellaneous Proofs to Know S. F. Ellermeyer Summer Semester 2010 Definition 1 An n n matrix, A, issaidtobeinvertible if there exists an n n matrix B such that AB BA I n (where I n is the

More information

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable)

Def. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable) Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November

More information

MA 527 first midterm review problems Hopefully final version as of October 2nd

MA 527 first midterm review problems Hopefully final version as of October 2nd MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes

More information

Even-Numbered Homework Solutions

Even-Numbered Homework Solutions -6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y

More information

Math 266: Phase Plane Portrait

Math 266: Phase Plane Portrait Math 266: Phase Plane Portrait Long Jin Purdue, Spring 2018 Review: Phase line for an autonomous equation For a single autonomous equation y = f (y) we used a phase line to illustrate the equilibrium solutions

More information

Homework Notes Week 6

Homework Notes Week 6 Homework Notes Week 6 Math 24 Spring 24 34#4b The sstem + 2 3 3 + 4 = 2 + 2 + 3 4 = 2 + 2 3 = is consistent To see this we put the matri 3 2 A b = 2 into reduced row echelon form Adding times the first

More information

Math 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( )

Math 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( ) #7. ( pts) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, λ 5 λ 7 t t ce The general solution is then : 5 7 c c c x( 0) c c 9 9 c+ c c t 5t 7 e + e A sketch of

More information

Solutions Problem Set 8 Math 240, Fall

Solutions Problem Set 8 Math 240, Fall Solutions Problem Set 8 Math 240, Fall 2012 5.6 T/F.2. True. If A is upper or lower diagonal, to make det(a λi) 0, we need product of the main diagonal elements of A λi to be 0, which means λ is one of

More information

7 Planar systems of linear ODE

7 Planar systems of linear ODE 7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

More information

Part II Problems and Solutions

Part II Problems and Solutions Problem 1: [Complex and repeated eigenvalues] (a) The population of long-tailed weasels and meadow voles on Nantucket Island has been studied by biologists They measure the populations relative to a baseline,

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

Math Ordinary Differential Equations

Math Ordinary Differential Equations Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x

More information

Math Matrix Algebra

Math Matrix Algebra Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,

More information

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. In this problem, if the statement is always true, circle T; otherwise, circle F. Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

More information

Find the general solution of the system y = Ay, where

Find the general solution of the system y = Ay, where Math Homework # March, 9..3. Find the general solution of the system y = Ay, where 5 Answer: The matrix A has characteristic polynomial p(λ = λ + 7λ + = λ + 3(λ +. Hence the eigenvalues are λ = 3and λ

More information

235 Final exam review questions

235 Final exam review questions 5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

More information

Department of Mathematics IIT Guwahati

Department of Mathematics IIT Guwahati Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,

More information

STABILITY. Phase portraits and local stability

STABILITY. Phase portraits and local stability MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),

More information

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016 Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the

More information

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling 1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples

More information

= e t sin 2t. s 2 2s + 5 (s 1) Solution: Using the derivative of LT formula we have

= e t sin 2t. s 2 2s + 5 (s 1) Solution: Using the derivative of LT formula we have Math 090 Midterm Exam Spring 07 S o l u t i o n s. Results of this problem will be used in other problems. Therefore do all calculations carefully and double check them. Find the inverse Laplace transform

More information

Math 308 Final Exam Practice Problems

Math 308 Final Exam Practice Problems Math 308 Final Exam Practice Problems This review should not be used as your sole source for preparation for the exam You should also re-work all examples given in lecture and all suggested homework problems

More information

Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form

Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form d x x 1 = A, where A = dt y y a11 a 12 a 21 a 22 Here the entries of the coefficient matrix

More information

+ i. cos(t) + 2 sin(t) + c 2.

+ i. cos(t) + 2 sin(t) + c 2. MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,

More information

MATH 1553 PRACTICE MIDTERM 3 (VERSION B)

MATH 1553 PRACTICE MIDTERM 3 (VERSION B) MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.

More information

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors. Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

More information

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2)

ENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2) ENGI 940 4.06 - Linear Approximation () Page 4. 4.06 Linear Approximation to a System of Non-Linear ODEs () From sections 4.0 and 4.0, the non-linear system dx dy = x = P( x, y), = y = Q( x, y) () with

More information

Section 9.3 Phase Plane Portraits (for Planar Systems)

Section 9.3 Phase Plane Portraits (for Planar Systems) Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable

More information

Vectors, matrices, eigenvalues and eigenvectors

Vectors, matrices, eigenvalues and eigenvectors Vectors, matrices, eigenvalues and eigenvectors 1 ( ) ( ) ( ) Scaling a vector: 0.5V 2 0.5 2 1 = 0.5 = = 1 0.5 1 0.5 ( ) ( ) ( ) ( ) Adding two vectors: V + W 2 1 2 + 1 3 = + = = 1 3 1 + 3 4 ( ) ( ) a

More information

Solution Set 7, Fall '12

Solution Set 7, Fall '12 Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det

More information

Problem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form

Problem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form Problem set 6 Math 207A, Fall 2011 s 1 A two-dimensional gradient sstem has the form x t = W (x,, x t = W (x, where W (x, is a given function (a If W is a quadratic function W (x, = 1 2 ax2 + bx + 1 2

More information

Math 216 Second Midterm 28 March, 2013

Math 216 Second Midterm 28 March, 2013 Math 26 Second Midterm 28 March, 23 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 that

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

Two Dimensional Linear Systems of ODEs

Two Dimensional Linear Systems of ODEs 34 CHAPTER 3 Two Dimensional Linear Sstems of ODEs A first-der, autonomous, homogeneous linear sstem of two ODEs has the fm x t ax + b, t cx + d where a, b, c, d are real constants The matrix fm is 31

More information

Math 331 Homework Assignment Chapter 7 Page 1 of 9

Math 331 Homework Assignment Chapter 7 Page 1 of 9 Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a

More information

Question 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented

Question 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented Question. How many solutions does x 6 = 4 + i have Practice Problems 6 d) 5 Question. Which of the following is a cubed root of the complex number i. 6 e i arctan() e i(arctan() π) e i(arctan() π)/3 6

More information

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015 Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal

More information

Linear Algebra Exercises

Linear Algebra Exercises 9. 8.03 Linear Algebra Exercises 9A. Matrix Multiplication, Rank, Echelon Form 9A-. Which of the following matrices is in row-echelon form? 2 0 0 5 0 (i) (ii) (iii) (iv) 0 0 0 (v) [ 0 ] 0 0 0 0 0 0 0 9A-2.

More information

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

More information

ANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1

ANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1 ANSWERS Final Exam Math 50b, Section (Professor J. M. Cushing), 5 May 008 PART. (0 points) A bacterial population x grows exponentially according to the equation x 0 = rx, where r>0is the per unit rate

More information

Linear Algebra Review

Linear Algebra Review Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite

More information

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM 33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm

Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is

More information

Old Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University

Old Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall

More information

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.

Solutions to Dynamical Systems 2010 exam. Each question is worth 25 marks. Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the

More information

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

More information

Half of Final Exam Name: Practice Problems October 28, 2014

Half of Final Exam Name: Practice Problems October 28, 2014 Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half

More information

Solutions to Math 53 Math 53 Practice Final

Solutions to Math 53 Math 53 Practice Final Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points

More information

Math 21b. Review for Final Exam

Math 21b. Review for Final Exam Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a

More information

Problem set 7 Math 207A, Fall 2011 Solutions

Problem set 7 Math 207A, Fall 2011 Solutions Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase

More information

Eigenpairs and Diagonalizability Math 401, Spring 2010, Professor David Levermore

Eigenpairs and Diagonalizability Math 401, Spring 2010, Professor David Levermore Eigenpairs and Diagonalizability Math 40, Spring 200, Professor David Levermore Eigenpairs Let A be an n n matrix A number λ possibly complex even when A is real is an eigenvalue of A if there exists a

More information

Third In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix

Third In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.

More information

Math 320, spring 2011 before the first midterm

Math 320, spring 2011 before the first midterm Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them. Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

More information

8.1 Bifurcations of Equilibria

8.1 Bifurcations of Equilibria 1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations

More information

DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication.

DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication. DIAGONALIZABLE LINEAR SYSTEMS AND STABILITY. 1. Algebraic facts. We first recall two descriptions of matrix multiplication. Let A be n n, P be n r, given by its columns: P = [v 1 v 2... v r ], where the

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

More information

Math 21b Final Exam Thursday, May 15, 2003 Solutions

Math 21b Final Exam Thursday, May 15, 2003 Solutions Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in

More information

Matrices and Linear Algebra

Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

More information

Math 232, Final Test, 20 March 2007

Math 232, Final Test, 20 March 2007 Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.

More information

Answers and Hints to Review Questions for Test (a) Find the general solution to the linear system of differential equations Y = 2 ± 3i.

Answers and Hints to Review Questions for Test (a) Find the general solution to the linear system of differential equations Y = 2 ± 3i. Answers and Hints to Review Questions for Test 3 (a) Find the general solution to the linear system of differential equations [ dy 3 Y 3 [ (b) Find the specific solution that satisfies Y (0) = (c) What

More information

One Dimensional Dynamical Systems

One Dimensional Dynamical Systems 16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar

More information

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

Question: Total. Points:

Question: Total. Points: MATH 308 May 23, 2011 Final Exam Name: ID: Question: 1 2 3 4 5 6 7 8 9 Total Points: 0 20 20 20 20 20 20 20 20 160 Score: There are 9 problems on 9 pages in this exam (not counting the cover sheet). Make

More information

MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.

MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ

More information

Calculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y

Calculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control

Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/

More information

Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12

Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12 Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider

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

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Systems of Algebraic Equations and Systems of Differential Equations

Systems of Algebraic Equations and Systems of Differential Equations Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS

MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on

More information

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

20D - Homework Assignment 5

20D - Homework Assignment 5 Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D - Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A

More information

π 1 = tr(a), π n = ( 1) n det(a). In particular, when n = 2 one has

π 1 = tr(a), π n = ( 1) n det(a). In particular, when n = 2 one has Eigen Methods Math 246, Spring 2009, Professor David Levermore Eigenpairs Let A be a real n n matrix A number λ possibly complex is an eigenvalue of A if there exists a nonzero vector v possibly complex

More information

CAAM 335 Matrix Analysis

CAAM 335 Matrix Analysis CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points

More information

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below. 54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and

More information

There are six more problems on the next two pages

There are six more problems on the next two pages Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with

More information

Stability of Dynamical systems

Stability of Dynamical systems Stability of Dynamical systems Stability Isolated equilibria Classification of Isolated Equilibria Attractor and Repeller Almost linear systems Jacobian Matrix Stability Consider an autonomous system u

More information

Linear Algebra and ODEs review

Linear Algebra and ODEs review Linear Algebra and ODEs review Ania A Baetica September 9, 015 1 Linear Algebra 11 Eigenvalues and eigenvectors Consider the square matrix A R n n (v, λ are an (eigenvector, eigenvalue pair of matrix A

More information

9.1 Eigenanalysis I Eigenanalysis II Advanced Topics in Linear Algebra Kepler s laws

9.1 Eigenanalysis I Eigenanalysis II Advanced Topics in Linear Algebra Kepler s laws Chapter 9 Eigenanalysis Contents 9. Eigenanalysis I.................. 49 9.2 Eigenanalysis II................. 5 9.3 Advanced Topics in Linear Algebra..... 522 9.4 Kepler s laws................... 537

More information

Linear Algebra - Part II

Linear Algebra - Part II Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T

More information

2018 Fall 2210Q Section 013 Midterm Exam II Solution

2018 Fall 2210Q Section 013 Midterm Exam II Solution 08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

More information

Part IA. Vectors and Matrices. Year

Part IA. Vectors and Matrices. Year Part IA Vectors and Matrices Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2018 Paper 1, Section I 1C Vectors and Matrices For z, w C define the principal value of z w. State de Moivre s

More information

Homogeneous Constant Matrix Systems, Part II

Homogeneous Constant Matrix Systems, Part II 4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

ENGI Duffing s Equation Page 4.65

ENGI Duffing s Equation Page 4.65 ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing

More information

Math 118, Fall 2014 Final Exam

Math 118, Fall 2014 Final Exam Math 8, Fall 4 Final Exam True or false Please circle your choice; no explanation is necessary True There is a linear transformation T such that T e ) = e and T e ) = e Solution Since T is linear, if T

More information

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre

More information

MATH 235. Final ANSWERS May 5, 2015

MATH 235. Final ANSWERS May 5, 2015 MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

More information

Differential Topology Final Exam With Solutions

Differential Topology Final Exam With Solutions Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function

More information