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Math 211 Second Midterm Exam: Solutions and Exercises, Exams of Differential Equations

Rice UniversityDifferential Equations

A comprehensive set of exercises and solutions for a second midterm exam in math 211, covering topics such as linear algebra, differential equations, and vector spaces. A valuable resource for students preparing for similar exams, offering detailed explanations and worked-out solutions to various problems. It is particularly useful for students seeking to solidify their understanding of key concepts and practice problem-solving techniques.

Typology: Exams

2017/2018

Uploaded on 03/30/2025

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Math 211–Spring 2018
Second Midterm Exam
This exam has 8 problems worth 126 points distributed over 12 pages, including this one.
Instructions: This is a 2 hour exam. You may not consult any notes or books during
the exam, and no calculators are allowed. Show all of your work on each problem and
carefully justify all answers. Points will be deducted for irrelevant, incoherent or incorrect
statements, and no points will be awarded for illegible work. If you run out of room, you
may work answers on the back of pages or on attached scratch paper. Be sure to clearly
indicate where work is continued on another page.
Section (mark one):
Section 002 Section 006 Section 007 Section 008
S. Sukhtaiev B. Orcan-Ekmekci C. Douglas S. Li
MWF 11am MWF 11am TTh9:25am MWF 2pm
Name:
Honor Pledge:
On my honor, I have neither given nor received any unauthorized aid on this exam.
Signature:
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Math 211 Second Midterm Exam: Solutions and Exercises and more Exams Differential Equations in PDF only on Docsity!

Math 211–Spring 2018

Second Midterm Exam

This exam has 8 problems worth 126 points distributed over 12 pages, including this one. Instructions:the exam, and no calculators are allowed. This is a 2 hour exam. You may not consult any notes or books during Show all of your work on each problem and carefully justify all answers. Points will be deducted for irrelevant, incoherent or incorrectstatements, and no points will be awarded for illegible work. If you run out of room, you may work answers on the back of pages or on attached scratch paper.indicate where work is continued on another page. Be sure to clearly

Section (mark one):

S. SukhtaievSection 002^ B. Orcan-EkmekciSection 006^ Section 007C. Douglas Section 008S. Li MWF 11am MWF 11am TTh9:25am MWF 2pm

Name:

Honor Pledge: On my honor, I have neither given nor received any unauthorized aid on this exam. Signature:

Question Points Score

Total: 126

  1. (9 points) Solve the initial value problem y′′^ + 8y = cos(3t), y(0) = 1, y′(0) = − 1.
  1. (28 points) This problem consists of three parts.(a) (8 points) Find the general solution of the differential equation y′′^ − 2 y′^ + y = sin x.

(b) (12 points) Find the general solution of the differential equation y′′^ − 2 y′^ + y = e xx 2.

  1. (15 points) Consider the differential equation y′′^ + k^2 y = ekx. (a) For every k ∈ R, find the general solution of the differential equation.

(b) Find all k for which the general solution (from part (a)) goes to +∞ as x → +∞.

  1. (27 points)the real line then these functions must be linearly dependent in (a) (3 points) If the Wronskian of n functions vanishes at some point on R. True False (b) (3 points) Let y′′ (^) + 7y′ (^) + 3y = 0, and lety 1 , y 2 be linearly independent solutions of the differential equation W [y 1 , y 2 ] be the Wronskian of these solutions. Then WTrue y 1 , y 2 False 6 = 0. (c) (3 points) Letand let W [y 1 , y y 2 , y 1 , y 3 ] be the Wronskian of these solutions. Then 2 , y 3 be solutions of the differential equation W y ′′y^1 + 7, y 2 y, y′^ + 3 3 = 0.y = 0, True False (d) (3 points) If the Wronskian ofthen these functions must be linearly dependent in n functions vanishes at all points on the real line R. True False (e) (3 points) There exist four vectors True False v~ 1 , ~v 2 , ~v 3 , ~v 4 that form a basis of R^3 (f) (3 points) There exist vectors True False v~ 1 , ~v 2 , ~v 3 , ~v 4 such that span{ v~ 1 , ~v 2 , ~v 3 , ~v 4 } = R^3 (g) (3 points) There exist vectors True False w~ 1 , ~w 2 , ~w 3 such that span{ w~ 1 , ~w 2 , ~w 3 } = R^4 (h) (3 points) dim

 a + b + c bc

: a, b, c ∈ R

(i) (3 points) Consider a set of functionsTrue^ False Then S is a vector space.^ S^ :=^ {f^ :^ R^ →^ R^ |^ f^ is differentiable and^ f^ (0) +^ f^ ′(0) = 1}. True False

  1. (15 points) Match the differential equations with the graphs of their solutions Differential EquationGraph^ (A)^ (B)^ (C)^ (D)^ (E) (1) y′′^ + y′^ + y = 0 (2) y′′^ − 3 y′^ − 4 y = 0 (3) y′′^ + y = 0, (4) y′′^ = 0 (5) y′′^ − 3 y′^ + 4y = 0

(A)

(B)

TURN PAGE TO SEE (C), (D) and (E)

(C)

(D)

(E)