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# Math 340 - Week 8 Notes Math 340

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This 7 page Class Notes was uploaded by Susan Ossareh on Wednesday March 16, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 18 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 03/16/16

Math 340 Lecture Introduction to Ordinary Differential Equations March 7 2016 What We Covered 1 Course Content Chapter 7 Matrix Algebra a Section 77 Determinants i What is the point of finding the determinant 1 Essentially it gives us an easy way to tell if a matrix has a trivial nullspace It can also easily tell us when a square matrix is singular or nonsingular we sort of solved for this in section 76 but that method is generally inadequate ii Recall 1 ad be is the determinant of matrix A 2 de nition let Amm aij the determinant of A is detm Z10a101 azaz awn 039 a where the sum is over all permutations of the rst 11 integers b This essentially means there are n terms in the definition of the determinant if n is 5 or larger the number of terms is basically unacceptable for the purposes of computation So we will actually never use this definition to compute the determinant iii Odd and Even 1 o is even if it can be made into its identity with an even number of changes If not it is odd 0 1 ifoisodd 1 1 if a is even 2 Given a square matrix you can take a take an ordered list of integers Any combination of that order can be rearranged back to the original combination The number of interchanges to get back to the original order de nes whether that permutation is even or odd iv Example 11 12 13 A3963 21 22 23 31 32 33 1 Permutations on 1 2 3 01 1 2 3 gt 1 2 3 this is even because it took zero interchanges for the permutation to revert back to the original order 022 1 3 gt 1 and 2 interchange gt 1 2 3 This is odd because it took 1 interchange for the permutation to revert back to the original order 031 3 2 gtinterchange3and2 gt 1 2 30dd 043 2 1 gtinterchange land 3 gt 1 2 30dd 05 3 1 2 gt interchange 1 and 2 gt interchange 1 and 3 gt 1 2 3 even a6 2 3 1 gt interchange 1 and 2 gt interchange 2 and 3 gt 1 2 3 even V Theorem If Anxn is such that detA isn t zero then A is nonsingular and vice versa vi Determinants and Row Operations 1 Proposition the determinant of a triangular matrix is the product of the diagonal entries 2 Anxn is triangular if it looks like 0 0 O Where the zeros in the top right corner of the matrix makes a triangle This was one example a triangle matrix is any matrix with a triangle of zeros in any one of its corners 3 We want to know how row operations on matrices affect the determinant in order to make good use of the proposition vii Example 1 1 3 A O 2 15 O O 4 detA 1 2X4 8 viii Proposition Let A be a nxn matrix 1 If the matrix B is obtained from A by adding a multiple of one row column to another then detB detA 2 If the matrix B is obtained from A by interchanging two rows or columns then detB detA 3 If the matrix B is obtained from A by multiplying a row by a constant c then detB c detA ix Example Compute the determinant of A where the columns are C1 C2 and C3 C1C3 gtC3 1 O O 2 1 2 3 3 5 5 2C2 C3 gtC3 1 O 2 1 O 3 3 F detB detA 11 detA 31 detA Suggested Homework 0 Section 77 24 26 32 42 44 Math 340 Lab Introduction to Ordinary Differential Equations March 8th 2016 What We Covered 1 Worksheet 7 a Highlights i We went over it in class as an introduction to section 81 2 Course Content Chapter 7 Matrix Algebra a Section 77 Determinants continued i Recall 1 If we have a one by two matrix of vectors where A12 171 v2 we can suppose the vectors are 121 Z and v2 Therefore a c A b d We know the determinant to be ad bc but in the context of vectors the determinant nds the area plane between the two vectors In a three dimensional realm we can have three vectors which are all linearly dependent when the determinant is zero ii Proposition Let the collection of vectors v1 v2 12 E IRquot then 121 12 are linearly independent if and only if the detA doesn t equal zero and where the A1X2 matrix exists iii Expansion of a determinant by a row or a column 1 We can nd the determinant of and nxn matrix of a de ned ij minor by a detA Zy1 1ijaij detAlj for anyi b detA 1 1ijajdetAj for any j iv Example Compute the determinant of 3 5 O A 2 3 4 1 6 1 1 Hint Pick the column or row with the most zeros so it s easier to compute In this case pick column 3 So we will use the equation for detA for any i and use each element in column three to solve detA 113013det2 3 1 6 123 423det1 z 133 133det 53 O 1 418 5 1 1 9 10 71 v Example Compute the determinant of 1 2 2 A 3 2 1 4 O 5 1 So here we ll pick row 3 because of the zero we could pick column 2 but the idea here is to show an example of the other determinant formula for anyi detA 131431det22 12 132032det 12 133533det 22 4X2 1 0 5 8 48 vi Proposition If A has 2 columns or rows which are equal or factors ratios of each other then the determinant of A is zero 1 Example 1 11 9 B 3 4 5 1 11 9 detB 0 vii Differences between nonsingular and singular for nxn A NONSINGULAR A SINGULAR AXB IS ALWAYS CONSISTENT Axb might be inconsistent AXo IS A TRIVIAL SOLUTION Axo has a nontrivial solution NULLA o NullA is nontrivial A IS INVERTIBLE IF DETA ISN T 0 A isn t invertible if detA is zero ROWS AND COLUMNS OF A ARE LINEARLY Rows and columns of A are linearly dependent INDEPENDENT Suggested Homework 0 Section 77 24 26 32 42 44 o Worksheet 7 0 Study for quiz tomorrow on section 77 Math 340 Lecture Introduction to Ordinary Differential Equations March 9th 2016 What We Covered 1 Worksheet 8 a Highlights i Went over it in class 2 Course Content Chapter 8 An Introduction to Systems a Section 81 De nitions and Examples i System of Differential Equations a set of one or more equations involving one or more unknown functions ii SIR Model of an epidemic 1 We can use SIR to model the effect a disease has on population N We can assume a few things then a The disease doesn t last long and isn t fatal b It spreads by contact between individuals c Individuals who recover become immune 2 SIR actually stands for a St susceptible b It infected c Rt recovered 1 And the population is N S I R 3 In a SIR model we can conclude that we have a system of 3 differential equations which are nonlinear and autonomous S aSI I a5 b R M iii Initial Value Problem 1 Through a biological stand point the populations don t start at 0 when analyzing an epidemic which means all three populations must have a starting point or an initial value if you will Sto 50 1to 10 Rto R0 2 So we will get the initial value problem 7139 fuWith ut0 uo SOIIOIR0T iv First Order Systems 1 A general first order system looks like 96 ftxy y gtxy x39t f txt3It 3quott 9txtYt 2 We will consider systems of n equations with n unknowns the number of equations must equal the number of dimensions of the system If the unknown functions are x1 t x2 t and xn t then the system has the form 751 f1tx1rxn X39z f2trx11rxn 7511 fntr x1 r xn 3 The solution is a vector Where the xit s satisfy the system 751 t xn t v Vector Notation 1 With systems you can reassign names to functions For instance the generic SIR model can have the unknown functions relabeled as the following u1t St u2t 1t u3t Rt u 1 au1u2 u z auluz buz u 3 bu2 Suggested Homework 0 Section 81 4 8 14 16 o Worksheet 8 Math 340 Lecture Introduction to Ordinary Differential Equations March 11th 2016 What We Covered 1 Worksheet 8 a Highlights i Went over in class 2 Course Content Chapter 8 An Introduction to Systems a Section 83 Qualitative Analysis i Existence and Uniqueness 1 Existence and uniqueness in this chapter is similar as what we went over in chapter 2 however it discussed differential equations in higher dimensionality Theorem Suppose the function ft x is defined and continuous in the region R and that the first partial derivatives of f are also continuous in R Then given any point to x0 for all real numbers the initial value problem x f t x with xt0 x0 has a unique solution defined in an interval containing to Furthermore the solution will be defined at least until the solution curve leaves the region R a We can assume R I U where I is the interval and U is the rectangle in R ii Uniqueness in Phase Space 1 3 We don t want to just copy chapter 2 though phase spaces bring new cards to the table through solution curves Geometric Fact two solution curves in phase space for an autonomous system cannot meet at a point unless the curves coincide As for intervals of existence solutions to linear systems with constant coefficients exist on all of R iii Equilibrium Points and Solutions 1 2 Example a Given the autonomous equation x f x b The equilibrium point is then x0 f x0 0 c And then the equilibrium solution is xt x0 Example x 1 x yx y 4 2x 7yy 092 11 x yx0 24 2x 7yy0 For equation 1 x 0 1 x y0 a 1xy a y1 x Y1 x can be recognized now as a linear equation For equation 2 y 0 4 2x 7 42x7y gt y a The lines are called nullclines b Where equation 1 and 2 intersect are equilibrium points so in this case there are 4 4 32 00T 07T 10T E E T c Existence and uniqueness work the same it s just more generalized Suggested Homework 0 Section 83 4 6 10 11a andb

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