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# Week 10 Notes Math 340

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This 5 page Class Notes was uploaded by Susan Ossareh on Sunday April 3, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 12 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 04/03/16

Math 340 Lecture – Introduction to Ordinary Differential Equations – March 28 , 2016 th What We Covered: 1. Worksheet 9 a. Highlights i. We went over a lot of the problems in class but the rest of the work sheet was homework so I won’t be giving the answers on here 2. However, the general way to solve the problems on the work sheet and for the upcoming quiz is: a. Assuming you have the equation ???? ???? = ???????? where A is a square matrix i. First, find the eigenvalue ???? ???? = (−1) det ???? − ???????? = 0 ii. From there we can find the eigenvector by ????????????????(???? − ????????) b. We can then format the final solution exponentially: ???? ???? = ???? ???? ???????? i. The general solution would then be something along the lines of… ???????? ???????? ???? ???? = ???? 1 ???? +.1.+ ???? ???? ???????? ???? Suggested Homework: Finish Worksheet 9 Math 340 Lab – Introduction to Ordinary Differential Equations – March 29 , 2016 th What We Covered: 1. Announcements a. Exam 2 next week, make sure you’re comfortable with all the material from now b. Quiz tomorrow i. covers topics 8.5-9.1 2. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.2: Planar Systems i. Planar Systems- so back in section 9.1 we were working with 1 dimensional systems, planar systems introduce us to 2D and we can, in turn, apply this method to higher dimensions ii. Recall: (???? ????) → (???? − ???? ???? ) ???? ???? ???? ???? − ???? ???? ???? = ???? − ???? ???? − ???? − ???????? ???? ???? = ???? − ???? ???? + ???? + (???????? − ????????); where (a+d) is the trace of A and (ad- cb) is the determinant iii. Complex Eigenvalues 1. Theorem: Suppose that A is a 2x2 matrix with complex conjugate eigenvalues ???? ???????????? ????. Suppose that w is an eigenvector associated with ????. Then the general solution to the system y’=Ay is ???? ???? = ???? ???? ???? + ???? ???? ???? ????????̅, where ???? and ???? are arbitrary constants 1 2 1 2 2. Proposition: Suppose that A is still a square matrix with real coefficients, and suppose that ???? ???? = ???? ???? + ???????? ???? is a solution to the system ???? = ???????? a. The complex conjugate ????̅ = ???? − ???????? is also a solution b. The real and imaginary parts x and y are also solutions, furthermore, if z and ????̅ are linearly independent, so are x and y 3. Euler’s Formula: this formula is key in solving for complex eigenvalues ???????? ( ) ???? = cos ???? + ????????????????(????) ???? = ???? + ????????; ????,????????ℝ ???? ????????= ???? (????+???????? ????= ???????????? ∗ ????????????cos ???????? ???????? = ???? cos ???????? + ???????? sin(????????) **Where the real part is the first element of the equation and the imaginary part is the second element of the equation** a. Example: Find the real solution i. Step One: pick an imaginary solution ii. “Split” the format into separate real and imaginary parts b. Foil: now foiling complex matrices isn’t actually foiling in the mathematical sense that we understand it to be, it’s more like simplifying by combining like terms i. Example: ???? [cos ???? )1 − icos t)0 + ???????????????? ????)1 − sin(????) 0 1 −1 1 −1 = ???? c[s ????( )1 − sin ????) 0 ] 1 −1 + ???????? c[s ????( ) 0 + sin ????)1] −1 1 cos ????) ????1???? = ???? (???? ), cos ???? + sin ???? ) ???? sin ????) ????2???? = ???? ( cos ???? + sin ???? ) 1. This is a fundamental set of solutions 2. The general solution is essentially the same as the f.s.s. however it also includes the constant c Suggested Problems: Section 9.2: 10, 16, 22, 30, 36, 44, 52 Math 340 Lecture – Introduction to Ordinary Differential Equations – March 30 , 2016 What We Covered: 1. Worksheet 9 a. Highlights i. Continued to work on problems as homework and in class 2. Quiz today a. Highlights i. Covered sections 8.5-9.1 3. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.4: The Trace-Determinant Plane i. What we can always assume from the generic equation ???? = ???????? is that all the solutions are linearly independent ii. Based off of that, we should expect a solution in the forma???? ???????? ???? ???? ???? iii. Example: 1 1 ???? = (−1 3 ) 2 ???? ???? = ???? − ???????? + ???? ???? ???? = ???? − 4???? + 4 = (???? − 2)2 → ???? = 2 ∗∗ ????ℎ???? ???????????????????? ????ℎ???? ???????????????????????????????? ???????? ???????????????????????? ???????? ???????? ????ℎ???? ???????????????????????????????????? ???????????????????????????????????????????????? ∗∗ −1 1 −1 1 ???? − 2???? = (−1 1 ) → ( 0 0 ) **The second column has 1 free variable which means it has 1 dimension so, in turn, only 1 geometric multiplicity** (???? 1) = ????( ) ???? 2 1 2???? 1 ????1???? = ???? (1) iv. Theorem: To help us see the linearly independency in solutions, we have the square matrix A with the representative eigenvalue and the eigenvectors 1 and v2, such that ???? − ???????? 2 = ????1then ???? 1 = ???? ???? an1 ???????? ????2???? = ???? (???? +2???????? ) 1 Suggested Homework: Be sure to be studying for the exam this coming week Section 9.4: 2, 4, 6, 18 st Math 340 Lecture – Introduction to Ordinary Differential Equations – April 1 , 2016 What We Covered: 1. Worksheet 9 a. Highlights i. Stillllllll going through the worksheet as classwork and homework 2. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Definitions: i. Eigenvalues: ???? ,....,???? 1 ???? ii. Algebraic Multiplicity: 1 ,...,???? iii. Geometric Multiplicity: ????1,...,???????? iv. ???? = ???? 1...,???? ???? ???? +1..+ ???? ???? v. Generalized eigenvectors: v is a generalized eigenvector of A, associated to the eigenvalue ???? if there exists ???????????? ???? 1. Essentially it is this: (???? − ????????) ???? = 0 vi. Exponential of a matrix 1 1 ∞ 1 ???? = ???? + ???? + ???? + ???? + ⋯ = ∑ ???? = ???????????????????????? 2! 3! ????=0????! vii. Example: viii. Proposition: 1. ???? ???? ????????= ???????? ???????? ???????? 2. ???? ???? = ???? ???? is a solution to ???? ???? = ???????? ???? , ???? 0 = ???? ???????? 3. Given ???? ???? = ???? ???? a. Left Hand Side: ???? ???? = ???????? ???? ???????? b. Right Hand Side: ???????? ???? = ???????? ???? ???????? ???????? ix. Essentially, your goal is to compute ???? ???? 1 ???? ???? = (???? + ???????? + ???? ???? +...)???? 2! 1 2 2 = ???????? + ???????????? + 2! ???? ???? ????+... 1 ????ℎ???????? ???????????? ???????? ???????????????????????????????????? ????????: ???? ????(????????) 2! x. Proposition: 1. If ???????? = 0 then ???? ???? = ???? 2. If ???? ???? = 0 3. THIS EQUATION IS REALLY REALLY IMPORTANT ????−1 ???????? 1 2 2 ???? ????−1 ???? ???? = ???? + ???????? + 2! ???? ???? +...+ (???? − 1 ! ???? xi. Proposition: 1. ???? ????+???? = ???? ???? ,???????? ???????????? ???????????????? ???????? ???????? = ???????? 2. If A is nonsingular then e is nonsingular and the inverse is e-A xii. Consider the matrix ???? − ????????, where ???? is an eigenvalue for A ????????(????−????????)= ???? ????????−????????????= ???? ???? −???????????? ???????? ???????????? ????(????−????????) ???? = ???? ???? xiii. Theorem: 1. [???? − ???????? ???? = 0 ???? ???? = ???? ???????????? ????(????−???????????? ???????? ???????? ???? ???? = ???? ???? 2. [???? − ????????] ???? = 0 ???????? ???????? ????(????−????????) ???? = ???? ???? ???? ???????? ???????? ????????−1 ????−1 ???? = ???? [???? + ???? ???? − ???????? ????+...+ (???? − ????????) ???? (???? − 1 ! xiv. Recall for the worksheet: 1. ???? ???? = ???? ???? is a solution to ???? ???? = ????????′ and ???? 0 = ????) ′ 2. To solve ???? ???? = ????????′ a. Find eigenvalues b. Find eigenvectors ′ ???????? c. Construct an exponential solution ???? ???? = ???? ???? ???? i. If: the number of solutions matches the number of dimensions then you’re good! ii. If not: the generic eigenvectors for the e.values with geometric multiplicity should be less than algebraic multiplicity Suggested Homework: Study for exam 2 Finish worksheet 9

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