×

Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

Math 340 Notes - Week 11

by: Susan Ossareh

9

0

4

Math 340 Notes - Week 11 Math 340

Marketplace > Colorado State University > Math > Math 340 > Math 340 Notes Week 11
Susan Ossareh
CSU

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

Covers the last few sections for chapter 9 and 4.3
COURSE
Intro-Ordinary Differen Equatn
PROF.
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Differential Equations
KARMA
25 ?

Popular in Math

This 4 page Class Notes was uploaded by Susan Ossareh on Tuesday April 12, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 9 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

×

Reviews for Math 340 Notes - Week 11

×

×

What is Karma?

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 04/12/16
MATH 340 – INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS What We Covered: April 4 1. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.6: The Exponential of a Matrix ???? 1 2 i. The exponential of the matrix A is defined to be ???? = ????2! + + 1 3 ∞ 1 ???? 3!???? +...= ∑????=0????!???? ii. Proposition: Suppose A is an nxn matrix 1. Then ???? ???????????? = ???????????????? ???? ???????? ???????? 2. If ???? ∈ ???? , the function ???? ???? = ???? ???? is the solution to the initial value problem ′ ???? = ????????,????????????ℎ ???? 0 = ???? iii. Proposition: Suppose A is an nxn matrix and v is an n-vector 1. If ???????? = 0,????ℎ???????? ???? ???? = ???? ???????????? ???????????? ???? 2 ???????? 2. If ???? ???? = 0,????ℎ???????? ???? ???? = ???? + ???????????? ???????????? ???????????? ???? 3. More generally, if ???? ???? = 0,????ℎ???????? ???? ???? = ???? + ????−1 ????????????+...+ ???? ????????−1???? ???????????? ???????????? ???? ????−1 ! iv. Example: Consider −4 −2 1 −1 0 ???? = ( ), ???? = ( ), ???????????? ???? = ( ) 4 2 −2 2 0 8 4 0 0 1 1. So let’s compute ???? ???? and ???? ???? a. We can say that −4 −2 1 −1 0 ???????? = ( 4 2 −2 )( 2 ) = (0) 8 4 0 0 0 b. Therefore, by the proposition, ???? ???? = ????. On the other hand, −4 −2 1 0 1 ???????? = ( 4 2 −2)( 0 = ( −2 ) = −???? 8 4 0 1 0 c. So ???? ???? = 0. Therefore it fulfills the proposition ???????? ???? ???? = ???? + ???????????? = ???? − ???????? ???? = (−2???? ) 1 v. Proposition: 1. If A and B are nxn matrices, then = ???? ???? if and only if AB = BA 2. If A is an nxn matrix, then ???? is a nonsingular matrix whose inverse is ???? vi. Suppose A is an nxn matrix, ???? is a number, and v is an n-vector 1. If [A - ????????]???? = 0,????ℎ???????? ???? ???? = ???? ???? for all t MATH 340 – INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS 2 ???????? ???????? 2. If [???? − ????????] ???? = 0, then ???? ???? = ???? (???? + ???? ???? − ???????? ???? ???????????? ???????????? ???? 3. More generally, if k is a positive integer and [???? − ????????] ???? = 0, then ????????−1 ???? ???? = ???? (???? + ???? ???? − ???????? ????+...+ ????−1 ![???? − ????????]????−1???? for all t b. Section 9.3 and 9.4 i. Solve 1 0 ???? = ( )???? 1???? −3 ???? ???? = (−1) det(???? − ????????) 1 − ???? 0 ( 1 −3 − ????) = 1 − ???? −3 − ???? − 1 0 = (1 − ????)(−3 − ????) ???? = 1 ???????????? ???? = −3 ????2= 4 ????1= 0 1 1 ii. General Solution 0 4 ???? ???? = ???? ????1 −3????( ) + ???? 2 ( ) 1 1 Suggested Homework:  Section 9.6: 2, 4, 8, 9, 13, 18, 20, 26, 30 What We Covered: April 5 1. Quiz a. Covers 9.2 and 9.5, 9.6 i. Find eigenvalues and eigenvectors ii. Real distinct, real repeated, complex iii. Find exp sol 1. ???? ???????? 2. ???? ????;???????????? ???????????????????????????????????????????? 2. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.3 and 9.4 i. Example: ???? = (1 0 ) ???? ???? = ???????? ???? ( ) ???? ???? = ???? ???? ( ) + ???? ???? −3????( ) 1 −3 1 1 2 1 ????1= 1 ????2= 0 MATH 340 – INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS 4 4???? ???? ???? ( ) ???? = ( ????) 1 ???? ???? ( ) 2???? ( ) 0.5???? ( ) ????−3????( ) −???? −3????( ) ???? ???? = ( ) 1 1 1 1 1 0 ′ 0 1. Left Hand Side: ???? ???? = ( ) 0 0 0 2. Right Hand Side: ???????? ???? = ????( ) 0 ( ) 0 3. This is the solution at the origin and it is an equilibrium solution ii. POSSIBLE EXAM QUESTION: Is this solution stable, unstable, or asym stable ′ ???? iii. Def: a solution x(t) for ???? ???? = ???? ????(????) is stable, unstable, asym stable if solutions with initial conditions close to it 1. They remain close as ???? → ∞ 2. They “go away” as ???? → ∞ 3. They converge to it as ???? → ∞ iv. Here, y(t) is unstable Suggested Homework:  Section 9.3: 2, 3, 10, 17, 20, 22 What We Covered: April 6 1. Quiz 2. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.7: Qualitative Analysis of Linear Systems ???????? i. Recall nxnthe solution of y’ = Ay are of the form ???? ???? = ???? ???? ????2 ???????? = ????????????(???? + ???????????? + ???? ????+...+ ???? ????) 2 ???? ii. Example: 2???? 2???? 1 0 2???? 1 ????2???? ???? ( 0 + ????( )1= ???? ( ???? ) = ( ???? ???? ) 1 1 1 + ???? ????2???? + ???? ???? 1. In general, the component of the solution of y’=Ay are of the form ???????? ???? ????(????) a. For complex solutions: ???? = ???? + ???????? the components look (????+???????? ???? l????????e ???? ????(????) b. ???? (???????????????????? + ???????????????????????? ????(????) iii. Theorem: A nxn with the eigenvalue (real or complex) MATH 340 – INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS 1. If real part of ???? > 0, then every solution goes to infinity as t goes to infinity 2. If real part ???? < 0, then the solution t goes to origin a t to infinity Suggested Homework:  Section 9.7: 2, 8, 10 What We Covered: April 8 1. Exam Sections: 4.1, 4.3, 7.7, 8.5, 8.3, 9.1-9.8 2. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.8: Higher Order Linear Equations i. A linear equation of order n is of the form ( ) 1. ???? (????)+ ???? 1 ????) ????−1 +...+ ???? ????−1 (???? ???? + ???? ???? ???? = ????(????) ii. Suppose the functions ???? ???? ,???? ???? ,...,???????????? ???? (????) are all defined on the 1 2 ???? interval ????,???? . The functions are linearly dependent if there are constants ????1,????2,...,???????????? ????????, not all them equal to , such tha1 1 ???? ???? + ( ) ( ) ????2 2???? + ...+ ???? ???????? ????= 0, for all ???? ???? (????,????). The functions are linearly independent if they are not linearly dependent. iii. Solution strategy: 1. Construct ????(????) 2. Solve ???? ???? = 0 ( ) ???????? 3. Find solution ???? ???? = ???? 3. Course Content – Chapter 4: Second-Order Equations a. Section 4.3: Linear, Homogeneous Equations with Constant Coefficients i. So what do you do with repeated roots? 1. If ???? has alg multi q then take ( ) ???????? ????1???? = ???? ????2???? = ???????? ???????? … ????−1 ???????? ???????????? = ???? ???? Suggested Homework:  Section 9.8: 14, 24, 28, 38  Section 4.3: 2, 10, 15, 18, 21, 26, 30, 35

×

25 Karma

×

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Janice Dongeun University of Washington

"I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

Refund Policy

STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com