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# Math 340 Notes - Week 11 Math 340

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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.

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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

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