dy dt = ty, y(0) = 3. (1)

Transcription

1 2. (10pts) Solve the given intial value problem (IVP): dy dt = ty, y(0) = 3. (1) 3. (10pts) A plot of f(y) =y(1 y)(2 y) of the right hand side of the differential equation dy/dt = f(y) is shown below. (a) Sketch a plot of the phase line. (b) Classify the equilibria as sources/sinks/nodes. (c) Sketch a plot of some solutions y(t) to the IVP dy/dt = f(y), y(0) = y 0 where y 0 is chosen to show the behavior of the solution on either side of the equilibria.? End of In Class Quiz 1. Out of Class Question on Next Page.?

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3 Last (Family) Name, First (Given) Name and U#: MA 231, Quiz 2 Directions: There is one question on this quiz (with parts on the front and back of this sheet) similar to the assigned HM. Provide your written solutions and answers to these quiz question(s) in the space provided on the front and back of this sheet. Don t forget to (print) your name and U# in the space provided. Illegibly written quizzes cannot be graded. Illegibly written names cannot have their score recorded. Good Luck. Prof. O. 1. (10pts) The DE dy/dt = y(1 y)(y 2) is hard to solve analytically. Two plots of Euler s method for two different initial conditions are shown. Figure 1: Actual MATLAB Euler s Method Output: (left) For IC u 0 =3, (right) For IC u 0 =4 (a) What step size is being used for each of the Euler s method runs? How many steps are taken? (b) For the plot on the left, u 1 =2.4 is the value of Euler s method after one step. Write down the Euler s method formula (with appropriate numbers) which gives rise to u 1 =2.4. Compute also u 2 and u 3.? More parts on back.?

4 (c) Give a short written explanation of why the numerical approximation on the left seems correct. (d) Give a short written explanation of why the numerical approximation on the right seems incorrect. What is a possible reason for the failure of Euler s method?? End of Quiz.?

5 2. (10pts) Answer the following questions about the initial value problem (a) Find the solution to the IVP. dy +4y = cos(3t), y(0) = 5. dt (b) What is the homogenous solution of this DE? What is a particular solution of this DE? (c) Sketch your solution in the (t,y)-plane of your solution paying particular attention to the steady state or equilibrium solution.? End of in class quiz. See next page for take home quiz problems.?

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7 Last (Family) Name, First (Given) Name and U#: MA 231, Quiz 4 Directions: There are two questions on this in class quiz (with parts on the front and back of this sheet) similar to the assigned HM. There is also a take home part of the quiz which is due Monday. Provide your written solutions and answers to these quiz question(s) in the space provided on the front and back of this sheet. Don t forget to (print) your name and U# in the space provided. Illegibly written quizzes cannot be graded. Illegibly written names cannot have their score recorded. Good Luck. Prof. O. 1. (6pts) Consider the linear system of DEs together with initial condition given by Ẏ = AY where A = and Y =. (1) 1 0 e 2t (a) Show that the functions Y 1 (t) = e t and Y 2 (t) = e 2t are solutions to the DE. (b) What are the eigenvalues and associated eigenvectors of the matrix A? (c) Solve the initial value problem.? One More Question on Back.?

8 (4pts) Is v 2A an eigenvector of the matrix A 3 2 7A? If so, find the eigenvalue (+2pts) If v is an eigenvector of A with eigenvalue,showthaty (t) =e t v is a solution to the differential equation Ẏ = AY. a b 4. (+2pts) A matrix of the form B = is called symmetic. ShowthatB has real (R) eigenvalues b d and that, if b 6= 0, thenbhastwodistinct (meaning, different) eigenvalues.? End of Quiz.?

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11 Last (Family) Name, First (Given) Name and U#: MA 231, Quiz 5 Directions: There are three questions on this in class quiz (with parts on the front and back of this sheet) similar to the assigned HM. Provide your written solutions and answers to these quiz question(s) in the space provided on the front and back of this sheet. Don t forget to (print) your name and U# in the space provided. Illegibly written quizzes cannot be graded. Illegibly written names cannot have their score recorded. Good Luck. Prof. O. 1. Find the solution to the first order system of linear differential equations dy dt = AY, Y (0) = Y 0 where A = and Y =. 5? One More Question on Next Page.? 1

12 2. Answer the following questions about the differential equation from problem 1: (a) What are the straight line solution(s) to the differential equation? (b) What is (are) the equilibrium solution(s)? (c) Characterize the equilibrium(s) as source/sink/saddle. (d) Sketch at least three phase plane solutions which illustrate the source/sink/saddle behavior of the equilibrium(s). Note: At least one of sketched solutions should be a straight line solution.? One More Question on Next Page.? 2

13 3. Find the solution to the first order system of linear differential equations dy 0 1 dt = AY, Y (0) = Y 0 where A = and Y = i given that one eigenvalue/eigenvector pair of A is ( 1,v 1 )= 3i, Note: Express your solution Y (t) as a vector with real (R) entries.? End of Quiz? 3

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17 MA 226: Exam 1B (100pts. total, 20pts each question) Last (Family) Name, First (Given) Name and U#: Directions: Provide your written solutions and answers to these exam questions in the space provided. Don t forget to (PrInT) your name and U# in the space provided. Illegibly written exams cannot be graded. If we cannot follow/understand your solutions, partial credit points cannot/will not be given. Exams with illegibly written names and U#s cannot have their score recorded. You will NoT need a calculator. You can now begin the exam. Good Luck. Prof. O. 1. Answer the following questions using the plot of the right-hand-side (rhs) of the differential equation = f(y) shown below: (DE) dy dt (a) What are the equilibrium solutions of this DE. (b) Sketch (1) aphaselineplotofthedeand then on the right plot (2) somesolutionsinthe (t,y)-plane which illustrate the behavior of each of the types of equilibria from part (b). Make sure you classify the equilibria as sources/sinks or nodes (c) Find a solution y(t) of the DE whose initial condition (IC) is y(0) = 0.? More Questions on Next Page.?

18 (a) (b) (c) (d) 2. Answer the following questions using the given slope fields. (a) Match the DEs below with their above slope field: dy dt dy dt = y t answer! answer = y sin(y)! dy dt = t2 answer! (b) Sketch the solution to the initial value problem (IVP) dy = y t, y( 1) = 1 On its appropriate dt slope field AbOvE for the maxiumum allowable time t. Clearly indicate the solutions initial condition.? More Questions on Next Page.?

19 3. Consider the differential equation dy dt = y2 2(t + 1)y +(t + 1) 2. (1) (a) Show that the functions y 1 (t) =t and y 2 (t) =t +2are solutions to eq. (1). Hint: Do not waste your time trying to find the general analytic solution. (b) Given that both f(t, y) and f y (t, y) are continous for all (t, y), what can you observe about another solution y 3 (t) whose initial condition is y 3 (0) = 1? Support you observation with a short explanation.? More Questions on Next Page.?

20 4. Answer the following questions about the family of differential equations, dy dt = f a(y) =y 3 + ay. (2) (a) Find the analytic solution to the IVP dy dt = f a=0(y), y(0) = 1. On what time interval does the analytic solution exist without "blow-ups"? (b) Sketch the bifurcation diagram for f a (y) from eq. (2) being sure to clearly label the bifurcation value(s). (c) Sketch some solutions y(t) to the DE dy dt = f a= 1 (y). Draw enough solutions which show the behavior near each of the equilibrium solutions.? More Questions on Next Page.?

21 5. You may find the following equation helpful on this problem, Z (t, y) = % dt where %=µ ( a(t)y b(t)). (3) Solve the differential equation, dy = dt 3y + 10 sin(t), (4) using potential function methods by answering the following questions. (a) Compute the integrating factor µ. (b) Find (t, y) =1, the c =1level curve of (t, y). What point (t, y) does the solution (t, y) =1 go through when t =0? (c) The potential function solution (t, y) =1is solvable for y. Whatistheexplicitsolutiony(t) to the DE (4)? What is the initial condition y 0 for this solution where y(0) = y 0?? More Questions on Next Page.?

22 (d) What is the steady-state part of the solution of y(t) and what is the transient part of the solution? (e) Give a rough sketch of your solution y(t) from part (c) which demonstrates the general behavior of the transient and steady state parts of the solution y(t). Also,giveashortexplanationofthis plot and how the DE (4) as a model relates to a thermocouple experiment. (f) (Bonus, no partial credit, +3pts), if you derive eq. (3) from scratch starting from the general form of a first order linear differential equation)

23 ? End of Exam. You may get up to turn in your exam. Please be considerate of your neighbor and try to be as quiet as possible on your way out. You should pick up your Lab problem after your turn in this exam.?

24 MA 226: Exam 2A (MWF) (100pts. total, 20pts each question) Last (Family) Name, First (Given) Name and U#: Discussion Section Directions: Provide your written solutions and answers to these exam questions in the space provided. Don t forget to (PrInT) your name and U# in the space provided. Illegibly written exams cannot be graded. If we cannot follow/understand your solutions, partial credit points cannot/will not be given. Exams with illegibly written names and U#s cannot have their score recorded. You will NoT need a calculator. You can now begin the exam. Good Luck. Prof. O. 1. Answer the following questions about eigenvalues and eigenvectors: (a) Is v = an eigenvector of the matrix A =? If so, find the eigenvalue (b) What are the eigenvalue(s) and associated eigenvector(s) of the matrix A = 1? 0 (c) You compute an eigenvalue and eigenvector pair of some matrix A to be 1 = 1 + i p p 5 and 1 i 5 v 1 =. 3 i. For the eigenvalue 1 =1+i p 2 5,yourfriendcomputesaneigenvectorw = 1+i p. 5 Give a very short explanation of why you can both be correct. Show how to convert from eigenvector v 1 to eigenvector w. ii. Find another eigenvalue/eigenvector pair ( 2,v 2 ) where 2 6= 1. 1

25 2. Match the plots below with the given information about the matrix A of dy dt = AY. Note: More than one plot may apply to each question. (a) A has complex eigenvalues of the form = ± i where < 0. ANS(s): (b) A has two real eigenvalues; both are positive. ANS(s): (c) A has two real eigenvalues; one is positive and the other is negative. ANS(s): (d) A has complex eigenvalues of the form = ±i. ANS(s): 2

26 3. For some matrix A assume that you know two solutions Y 1 (t) = Ẏ = AY. 2e 3t e 3t and Y 2 (t) = e t e t to (a) What are the eigenvalues of the matrix A? WhataretwoeigenvectorsofA? (b) Verify that Y (t) =Y 1 (t)+2y 2 (t) is a solution to Ẏ = AY. (c) In the phase plane, sketch the solutions Y 1 (t), Y 2 (t), andy (t). (Note: Label all parts of your plot and include arrows indicating the direction of motion along the solutions in positive time.) 3

27 4. Using eigenvalue and eigenvector methods, find the real solution to Ẏ = AY, Y (0) = Y 0 where A = and Y =. 1 4

28 5. Answer the following questions about the second order, linear differential equation mÿ + bẏ + ky =0 (1) where m>0,b > 0,k > 0 are the mass, friction, and spring constant of some mass spring damper system. The following definitions are made for computation and analysis ease, 2 = b m and!2 = k m. You may find the formula helpful: y(t) = 8 < : C 1 e ( +p 2! 2 )t + C 2 e ( p 2! 2 )t e t (C 1 cos( p! 2 2 t)+c 2 sin( p! 2 2 t) C 1 e t + C 2 te t (a) Assuming that Y = y v, what is the matrix A you get when you express eq. (1) as a linear system of differential equations? Note: Your matrix should be expressed in the variables and!. (b) Express the eigenvalues of your matrix A in terms of and!. (c) Given the values m =1,b =2,andk = 10 and the initial condition Y 0 = T of the solution y(t) to eq. (1)? 3 0, what is the period (d) Using the same values and initial condition as in (c), classify the equilibrium solution (y, v) =(0, 0) (underdamped, overdamped, critically damped, spiral sink, spiral source, sink,source, saddle) and give as accurate of a sketch as you are able to of the (t, y) and (y, v) plane plots. 5

29 3 (e) Using the same values and initial conditions as in (c), m =1,b =2,andk = 10 and Y 0 =, 0 y(t) find the solution Y (t) = to v(t) Ẏ = AY, Y (0) = Y 0. 6

30 6. (Bonus, 5pts, no partial credit) Find the solution to Ẏ = AY, Y 0 = where A =

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45 MA 226 Quiz 1A (5 pts. total) Last (Family) Name, First (Given) Name and U#: Directions: There is one questions on this quiz; similar to the assigned HM. Provide your written solutions and answers to these quiz question(s) in the space provided on the front and back of this sheet. Don t forget to (print) your name and U# in the space provided. Illegibly written quizzes cannot be graded. Illegibly written names cannot have their score recorded. You can begin the quiz, you have 20 mins. Good Luck. Prof. O. 1. Solve the given intial value problem: dy dt = ty, y(0) = 3 (1)? End of Quiz.?