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# Study guide 2 Math 340

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This 4 page Study Guide was uploaded by Susan Ossareh on Monday April 4, 2016. The Study Guide belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 10 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 04/04/16

Study Guide 2 Math 340 – Differential Equations Exam 2 – April 7 Disclaimer: This study guide will cover everything from section 7.6 to 9.4, however do not rely on this information to get you through the test, tis only an aide Section 7.6: Square Matrices Definitions: o The nxn matrix A is said to be nonsingular if we can solve the system Ax=b for any choice of the vector b in R . Otherwise the matrix is singular o An nxn matrix A is invertible if there is an nxn matrix B such that AB=I and BA = I. A matrix B with this property is called an inverse of A Propositions: o An nxn matrix A is nonsingular if and only if an equivalent matrix in row echelon form has only nonzero entries along the diagonal o If A is nonsingular, then the equation Ax=b has a unique solution x for every choice of the vector b o If A is an nxn matrix, then the homogeneous equation Ax=0 has a nonzero solution if and only if the matrix is singular o An nxn matrix A is invertible if and only if it is nonsingular Example: Is the matrix singular or nonsingular? −3 6 8 ???? = ( ) −1 2 1 0 0 1 1 3 − 3 1 ???? →2???? 2 5???? 2 ???? 3 ???? 3 −3 6 8 ( 5 ) 0 0 − ⁄ 3 0 0 0 ????ℎ???? ???????????????????????????????? ???????? ???????????????????? ???????? ????ℎ???? ???????????????????????????????? ???????????????????? ???????????? ???????????????????????????????? Section 7.7: Determinants Definitions: ???? o det ???? = ∑ ????(−1) ????1????1∗...∗????????????, where the sum is over all permutations of the first n integers Propositions: o The determinant of a triangular matrix is the product of the diagonal entries o Let A be an nxn matrix If the matrix B is obtained from A by adding a multiple of one row to another, then det(B)=det(A) If the matrix B is obtained from A by interchanging two rows, then det(B) = -det(A) If the matrix B is obtained from A by multiplying a row by a constant c, then det(B) = c det(A) o Let A be an nxn matrix ???? ????+???? For any i with 1 ≤ ???? ≤ ????, we have det ???? = ∑ ????=1−1) ???? ????????t(???? )???????? For any j wih 1 ≤ ???? ≤ ????, we have det ???? = ∑???? (−1)????+???????? det(???? ) ????=1 ???????? ???????? Suppose A and B are nxn matrices. Then det ???????? = ) det ???? det(????). This proposition has a nice corollary A collection of n vectors 1 , . .n. , x in R is a basis for R if and only if the matrix X=[x1x 2 x ],nwhose column vectors are the indicated vectors, has a nonzero determinant Section 8.1: Definitions and Examples Definitions: o Susceptible: those individuals who have never has the disease o Infected: those who are currently ill with the disease o Recovered: those who have had the disease and are now immune o Autonomous: the RHS doesn’t depend explicitly on the independent variable o Order: the highest derivative that occurs in the system o Dimension: number of unknowns o Planar: a system of 2 dimensions Section 8.2: Geometric Interpretation of Solutions Definitions: o Parametric Plot: plotting vector notation 2 o Phase Plane: a curve in R Section 8.3: Qualitative Analysis Definitions: o Nullcline: when the solution set is the union of two lines in the phase plane o Equilibrium Point: where both right-hand sides are equal to zero Page 02 o Equilibrium solution: constant functions based off of the equilibrium point Theorems: o Suppose the function ????(????,????) is defined and continuous in the region R and the first partial derivatives of f are also continuous in R. Then given any point, the initial value problem has a unique solution defined in an interval containing t0. Furthermore, the solution will be defined at least until the solution curve leaves the region o Suppose that A is a square matrix and f(t) is a column vector and that the components of both are continuous functions of t in an intervals. Then, ???? for any ????0∈ (????,????), and for any ???? 0 ℝ , the inhomogeneous system ′( ) ( ) ( ) ( ) ???? ???? = ???????? ???? + ???? ???? , with initial condition ???? ???? 0 = ????0, has a unique solution defined for all ???? ∈ (????,????) Section 8.4: Linear Systems Definitions: o Appear Linearly: there are no products, powers, or higher order functions involving the unknowns o Linear System: when the functions appear linearly Section 8.5: Properties of Linear Systems Definitions: o A set of k solutions to the linear system y’=Ay is linearly independent, then it is linearly independent for any one value of t Theorem: o Suppose x a1d x ar2 solutions to the homogenous linear system x’=Ax. If C1and C 2re any constants, then ???? = ???? ???? 1 1 ???? 2 2also a solution o Suppose that ???? ,1..,???? ????s also a solution. Thus for any constants ???? 1...,???????? the function ???? ???? = ???? ???? ???? + ???? ???? ???? +...+ ???? ???? ???? ???????? ???? ???????????????????????????????? ???????? ???? = ′ 1 1 2 2 ???? ???? ???????? Section 9.1: Overview of the Technique Definitions: o Suppose A is an nxn matrix. A number ???? is called an eigenvalue of A if there is a nonzero vector v such that Av=????????. If ???? is an eigenvalue, then any vector v satisfying this equation is called an eigen vector associated with the e.value Page 03 ???? o If A is an nxn matrix, the polynomial ???? ???? = (−1) det ???? − ???????? = det(???????? − ????) is called the characteristic polynomial of A, and the equations p(????) = ???? (−1) det ???? − ???????? = 0 is called the characteristic equation Propositions o The e.values of an nxn matrix A are the roots of its characteristicpolynomial o Let A be an nxn matrix and let ???? be an e.value of A. The set of all e.vectors associated with ???? is equal to the nullspace of A-????I. Hence the eigenspace of ???? is a subspace of R Page 04

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