Math 250B Midterm III Information Fall 2018

Transcription

1 Math 250B Midterm III Information Fall 2018 WHEN: Monday, December 10, in class (no notes, books, calculators I will supply a table of annihilators and integrals.) EXTRA OFFICE HOURS : Friday, December 7 5:30 PM 7:00 PM and 8:30 PM 12 MIDNIGHT Saturday, December 8 3:30 PM 5:00 PM Sunday, December 9 8:00 PM 12:00 MIDNIGHT Monday, December 10 11:00 AM 2:00 PM and 3:00 4:00 PM REVIEW SESSIONS: Saturday, December 8 from 1:00 3:30 PM in MH-565 Sunday, December 9 from 4:00 6:30 PM in MH-565 COVERAGE: The midterm will cover the material discussed in lecture from Sections 6.3, 6.4, , , 8.7, and STUDYING: Here is an overview of the topics we have covered: Chapter 6: Kernel and Range of a linear transformation, Generalized Rank-Nullity Theorem: dim[ker(t ) + dim[rng(t ) = dim[v, One-to-one, Onto, Isomorphism. Chapter 7: Eigenvalue, Eigenvector, Characteristic equation, Eigenspace corresponding to λ, Multiplicity of λ, Defective matrix, Nondefective matrix, Complete set of eigenvectors, Diagonalizable matrix. Chapter 8: Derivative operator, Linear differential operator of order n, General solution to nth order linear differential equation, Wronskian, Associated homogeneous differential equation, Complementary function, Auxiliary equation, Trial solution, Particular solution, Annihilator method, Variation-of-parameters method. Chapter 9: Linear system of differential equations: homogeneous vs. non-homogeneous, Vector/Matrix formulation of a system, Wronskian of vector functions, General solution to linear system, Particular solution to linear system, Generalized eigenvectors. THINGS TO BE ABLE TO DO:

2 Compute bases for Ker(T ) and Rng(T ), given a linear transformation T : V W. Utilize the equation dim[ker(t ) + dim[rng(t ) = dim[v to assist in analyzing Ker(T ) and Rng(T ). Determine whether a given linear transformation is one-to-one, onto, both, or neither. Determine whether an isomorphism is possible between two vector spaces V and W, and if so, be able to construct one. Understand that T : R n R n via T (x) = Ax for an n n matrix A is an isomorphism if and only if A is invertible. Be able to determine the eigenvalues of a matrix, including possibly complex eigenvalues. Be able to find corresponding eigenvectors for a given eigenvalue, and find a basis for the corresponding eigenspace. This requires you to find nullspace(a λi) for the eigenvalue λ. Be able to decide if a given matrix is defective or non-defective. eigenvalue pull its weight? Does each Be able to do basic, short proofs of theoretical results involving eigenvalues and eigenvectors (for example, Problems in Section 5.6, and Examples ). Be able to decide if a given matrix is diagonalizable. If so, be able to find an invertible matrix S and a diagonal matrix D such that S 1 AS = D. Be able to find the general solution to any n-th order linear homogeneous DE by finding roots of the characteristic equation. This also means you need to understand how to handle repeated roots and complex roots that may arise. If initial values are given, be able to solve an initial-value problem for a specific solution. Know the process for finding the general solution to an n-th order linear nonhomogeneous DE. How is the solution to the non-homogeneous DE related to the solution to the corresponding homogeneous DE?

3 Be able to use annihilators to find a particular solution to a non-homogeneous DE. Be able to apply the variation-of-parameters procedure to find a particular solution to a non-homogeneous DE. Be able to write down linear systems of differential equations, both homogeneous and non-homogeneous. Be able to solve a homogeneous linear system of differential equations with non-defective coefficient matrix. Be able to solve a homogeneous linear system of differential equations with defective coefficient matrix. ADVICE: I suggest reviewing the group work (starting with Group Work #10) and Quiz #3. You will not have time to re-do all of the homework, but you might try some of them again, especially the ones that will assist you with the THINGS TO BE ABLE TO DO listed above. TRY TO RE-DO PROBLEMS FROM SCRATCH, RATHER THAN JUST REVIEWING YOUR ALREADY-COMPLETED SOLU- TIONS. Also, quiz each other in study groups. Finally, you can ask me questions as much as you want, and I will be happy to review or pop-quiz a topic with you if you feel shaky. Basically, I m here to help and I want everybody to do well, so please don t be shy :=)!! Below is a list of some sample problems, in a random order, that may be similar to midterm questions. I will be posting solutions Practice Problems: Problem 1. Determine whether the following matrix is diagonalizable or not. If it is, find an invertible matrix S and a diagonal matrix D such that S 1 AS = D. A := Problem 2. Without a calculator, find A 10 if [ 0 1 A = 2 1.

4 Problem 3. (a): Write the DE y + 4xy 6x 2 y = x 2 sin x as an operator equation and give the associated homogeneous DE. (b): Write the DE (D 2 + 1)(D + 3)(y) = e 4x + ln(2x + 1) in terms of the expressions y, y, y, etc. Is this differential equation linear? Why or why not. Problem 4. Find the complementary function y c (x) and derive an appropriate trial solution for y p (x) for the given DE. Do not solve for the constants that arise in your trial solution. (a): (D + 1)(D 2 + 1)y = 4xe x. (b): (D 2 2D + 2) 3 (D 2) 2 (D + 4)y = e x cos x 3e 2x. Problem 5. Find a basis for the solution set of each differential equation below. In the case of part (b), use the Wronskian to verify that the basis solutions are indeed linearly independent. (a): (D 2 + 4) 2 (D + 1)y = 0. (b): y + 3y + 3y + y = 0. Problem 6. Write the general solution to each differential equation in Problem 5. Problem 7. Solve the IVP (D 3 + 2D 2 4D 8)y = 0 subject to the initial conditions y(0) = 0, y (0) = 6, y (0) = 8. Problem 8. Let v be an eigenvector of A corresponding to the eigenvalue λ. Prove that if B = S 1 AS, then S 1 v is an eigenvector of B corresponding to the eigenvalue λ. Problem 9. Find the general solution to the differential equation under the assumption that x > 0. y + 4y + 4y = e 2x x 2

5 Problem 10. Find the general solution to x (t) := x(t), and use the Wronskian to check your answer. Problem 11. (a): Verify that x p (t) := [ 3t 4 5t + 6 is a particular solution to the non-homogeneous system [ [ t 11 x (t) = x(t) (b): Find the general solution for the system. Problem 12. Juliet is in love with Romeo, who happens to be a fickle lover. The more Juliet loves him, the more he begins to dislike her. But when she dislikes him, his feelings for her warm up. On the other hand, her love for him grows when he loves her and withers when he dislikes her. A model for their ill-fated romance is dj dt = Ar and dr dt = Bj, where A and B are constants, r(t) represents Romeo s love for Juliet at time t, and j(t) represents Juliet s love for Romeo at time t. (a): Is A positive or negative? Why? What about B? (b): Derive a differential equation for r(t) which does not involve j(t) and solve it. (c): Find r(t) and j(t), in terms of A and B, given that r(0) = 1 and j(0) = 0. (d): As you may have discovered, the outcome of this relationship is a never-ending cycle of love and hate! Find what fraction of the time they both love one another. Problem 13. Let A be a square matrix. Find solutions of x = Ax of the form T (t)v, where T (t) is a scalar-valued function of t and v is an eigenvector of A. Problem 14. Ants are crawling around in Scott s apartment between a bottle of Von s Tea with a Twist in his kitchen and some leftover Domino s pizza in his

6 bedroom. Assume that each hour, one-third of the ants that were in the bedroom move to the kitchen, and one-fourth of the ants that were in the kitchen move to the bedroom. Let k(t) denote the number of ants in the kitchen at time t, and let b(t) denote the number of ants in the bedroom at time t. [ k(0) (a): If denotes the initial distribution of ants in the kitchen and bedroom b(0) (at time t = 0 hours), determine how many ants are in the kitchen and the bedroom after l hours. (b): As time approaches infinity, what fraction of the ants will be in each room? Does the initial distribution of ants affect the answer here? Why or why not? [ Problem 15. Solve the initial value problem x 0 = Ax, where x(0) = and 1 A = [