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# Ordinary Differential Equation MATH 2280

Weber State University

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This 14 page Class Notes was uploaded by Rosa Farrell on Wednesday October 28, 2015. The Class Notes belongs to MATH 2280 at Weber State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/230805/math-2280-weber-state-university in Math at Weber State University.

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Date Created: 10/28/15

SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 MIKE WILLS CONTENTS Introduction Second Order Equations and First Order Systems Vector Spaces Function Spaces FPON E oanH 1 INTRODUCTION The main purpose of this note is to ll in some of the details on the linear algebra side of things for our course For the most part7 you do not need to know this stuff for the class per 567 but perhaps it Will provide a bit of a motivation It is assumed that the reader is familiar With the matrix handouti A second purpose is to clarify some comments made in class 2 SECOND ORDER EQUATIONS AND FIRST ORDER SYSTEMS De nition 21 The most general second order ordinary differential equa tion is one that can be Written in the form 21 gt 7y7y7t 07 With 39i39 nonconstant With respect to Q so that the second derivative term does not just disappearI One can always Write a second order equation as a system of rst order equations by Writing 1 y39 and hence getting the system 39i39 i v y t 0 2 2 7 7 7 t v yr The above formulation is too general for our purposes The usual general formulation of a second order equation is 23 y fy397y7t and the equivalent rst order system is y39 1 2 4 l vfv7y7tA By de ning 2 5 Xy 2 MIKE WILLS and v 26 F lt gt l f l the system can be written in vector form as 27 X F Now F can be thought of as a function of X and t and that motivates the following de nition De nition 22 A rst order system of differential equations is one that can be written in the form 28 X FX t where t can be thought of as a time variable and X is a column vector of one or more time dependent variables7 11 12 In We will assume that there are n equations and so it is natural to refer to the system as n dimensional Remark 23 This is not the most general form7 but it is arguably the most typical Notice that the domain of F is in Rn1 and the codomain is R 39 One can add the extra equation t l and then rewrite the system as Y FY where Y is the column vector X with t adjoined at the end This will make the system autonomous In the case that X consists of only variable7 the system reduces to a single equation which we discussed in chapter 2 Theorem 24 Picard Suppose that F is continuous in a neighbourhood U of mmto Suppose that F11F 2FM are also continuous Then the initial value problem X FX t 29 XOO 1 0 has a unique solution near to Remark 25 The proof of the result starts by de ning 210 X1 X0t FX0sds z and inductively de ning O 211 Xn1 X0 EFXL7 sds to The integration of a vector quantity is understood to be the integration of each term in the vector Note that your correspondent has consistently got this wrong in class The rst term on the right hand side is X0 not Xn The proof works much the same way in any dimension For existence of a solution one merely requires that F be continuous for uniqueness7 Lipschitz continuity suffices Here are some examples of rst order systems in the two dimensional situation SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 3 Example 26 139 yr 212 y y t Example 27 139 7213 mm De nition 28 A rst order system is linear if there exists a matrix At and a column vector bt such that 214 XFXt AXb lf 1 is identically zero7 the system is homogeneous Otherwise7 the system is inhomogeneous As usual7 we will assume that the entries of A and b are continuous Recall that for any square matrix7 B7 00 En 215 expB eB Z W It can be shown that this matrix series always converges The general solution to the homogeneous system is 216 Xht exp t Asdsxo where X0 is an arbitrary vector A particular solution to the inhomogeneous equation can be found using the technique of variation of parameters write 217 Xpa exp t magma Plug Xp into the system and solve for W This will always work By the superposition principle7 the general solution to the inhomogeneous linear system is 218 X XthXp We now introduce some linear algebra that may make the theory discussed in chapter 3 a bit more palatab e 3 VECTOR SPACES We now summarize some of the key points from linear algebra that are discussed in distilled form in our text This is an overview7 and a somewhat sketchy one at that More details can be garnered from onels favorite linear algebra text De nition 31 Let V be a nonempty set Suppose that V is equipped with two operations7 Jr and such that the following hold for any uVW 6 V7 15 E R iiavaveV iii uvwuvw iv There exists 0 E V such that u0 0u u v uvvu 4 MIKE WILLS vi awv Mew amv vii a BV av BV viii aV W av aw ix 1 V V x There exists 7V E V such that V V 0 Then V is a real vector space Elements of V are vectors The operation is vector addition and the operation is scalar multiplication Remark 32 The above axiomatic characterization asserts that V is closed With re spect to vector addition and scalar multiplication that vector addition is associate commutative has an identity and additive inverses exist that scalar multiplication distributes over vector addition and that the multiplying a vector by one does not change the vector A complex vector space satis es the same axioms but in this situation a and B are arbitrary complex numbers More exotic vector spaces also exist but are of limited interest to us at this time In this handout all vector spaces are real unless noted otherwise Example 33 Our calculus textbook goes to great lengths to stress the difference between ndimensional vectors and coordinates in R This distinction is not observed much in linear algebra By identifying points and vectors we see that R and C are vector spaces Example 34 The set an of m X n matrices is a vector space under matrix addition De nition 35 Let V be a vector space Let U C V With U nonempty Suppose that U is closed under vector addition and scalar multiplication that is Whenever uV E U a E R it turns out that u V au 6 U Then U is a subspace of V Example 36 If V is a vector space then 0 and V are subspaces of V Example 37 The other subspaces of R2 are straight lines that pass through the origin Example 38 The other subspaces of R3 are lines and planes through the origin De nition 39 A set of n vectors S V1Vn in V is linearly indepen dent if n 31 Zuzka 0 ak Ofor all k k1 If S is not linearly independent it is said to be linearly dependent Example 310 The set 0 is linearly dependent Example 311 The set ij k C R3 is linearly independent De nition 312 The span of a set S of n vectors in V is the set n 32 W 04ka l 04k 6 k1 If W V then S spans V 1The latter is a complex Vector space SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 5 Proposition 313 The span of S is a subspace of W De nition 314 If S is a set of n linearly independent vectors in V and the span of S is V7 then S is a basis for V and V is an ndimensional vector space If there is no nite subset of V which spans V7 then V is in nite dimensional The vector space7 07 is zero dimensional Example 315 The vector space R is n dimensional The basis vectors form a basis Example 316 Straight lines that pass through the origin in R2 are one dimen sional subspaces of 2 Example 317 Planes that pass through the origin in R3 are two dimensional subspaces of R3 In the calculus sequence7 we do not deal with in nite dimensional vector spaces7 which are a bit tricky to handle However7 such spaces play a crucial role in analysis We will defer some examples until later in this handout De nition 318 Let V and W be vector spaces Let T V A W be a map such that for any u7 v E V and 15 E R the following equation holds 33 Tau v aTu BTW Then T is a linear operator or linear map from V to W Notation 319 One often writes Tu rather than Tu when T is a linear operator Example 320 If A is an m X n matrix7 then the map 11 gt gt A11 is a linear map from R to Rm As it happens7 these are the only linear operators that every one in math 2280 is likely to have seen at the start of the semester ln particular7 the only linear maps from R to R are those of the form y m1 De nition 321 If there exists a bijective linear map7 T7 between two vector spaces V and W then V and W are isomorphic and T is a isomorphism Example 322 The vector space7 V is isomorphic to itself under the identity map v gt gt v Example 323 One dimensional subspaces of R are isomorphic to each other and to R Example 324 If V and W are isomorphic and U and V are isomorphic7 then U and W are isomorphic Hence7 the relation is isomorphic to7 is an equivalence relation Example 325 If V and W are both ndimensional vector spaces7 then V and W are isomorphic Thus7 up to isomorphism7 there is only one n dimensional vector space Similar assertions can be made for vector spaces of any cardinality but that requires rather advanced set theory in particular7 the axiom of choice 6 MIKE WILLS 4 FUNCTION SPACES Many of the most useful vector spaces are what are called function spaces indeed the ones that we use in this course are exactly that What we mean y a function space is a set whose elements are functions equipped with suitable vector space operations Example 41 We de ne lP n to be the set of all polynomials with real coef cients in one variable of degree less than or equal to n together with the zero polynomial whose degree is unde ned or 700 depending on ones point of view For example 41 P1 mm a0 l a1a0 E R Then lP n is a vector space with p q 2 101 41 and ap 0471 A basis for P1 is the set 11 A basis for P4 is 141312zl Hence lP n is isomorphic to RWA The set of all polynomials is also a vector space this time of in nite dimension and is denoted lP Notice that lP O C P1 C P2 H C P Recalling that sequences are functions whose domain is N any vector space whose vectors are sequences can be thought of as a function space Example 42 The set S of sequences of real numbers is a vector space Given a anb 127 E S we de ne ab an bn and for a E R aa aan lmportant subspaces of S are the socalled little el p spaces For example 42 1aESlZlanl lt00 is the set of all absolutely summable sequences Also 43 lma68llanlltM is the set of all bounded sequences Here M depends on the sequence a but not on n Thus different sequences in loo have different M 7s For p 2 l 44 17 a e s i 2 law lt 00 Notice that 1 C loo for any p 2 l The set lg is notable since it has an inner product which is a generalization of the dot product from vector calculus Hence lg has a notion of orthogonality Example 43 Let U C Rm be a closed set De ne f y U A R17 l f is a function Then F is a vector space with y yt zt and ay t ayt for every y z E f t E U and a R We will primarily be concerned with subspaces of 7 When considering ODEs p will be equal to one When considering systems p will be the number of equations in the system Notice that if U R then P C 7 Probably the most important subspaces of f are the socalled big el p spaces LP To de ne those properly requires a bit of measure theory in particular Lebesgue integration which would be a bit of a distraction We therefore turn to the vector spaces of interest to us in ODEs SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 7 De nition 44 For n E N U 0 we de ne C U as follows 4 5 C U y E f l all nth order partial derivatives of y exist and are continuous We recall that the 0th derivative of a function is just the function itself Furt er7 46 CooU y E f l all partial derivatives of all orders of y exist and are continuous Notation 45 If the set U is immaterial or understood7 we will write C for C U We assume for the remainder of this handout that U C R is an interva Proposition 46 The sets C and C00 are in nitedimensional vector spaces Proof The reader should be able to verify this D Proposition 47 The map L C A 7 given by 47 Lltygt 2am k0 where ak U A R is an arbitrary function is a linear operator We will assume that each ak is continuous We will also assume that an is not the zero function Proposition 48 The set of solutions to the equation Ly 0 is an ndimensional subspace of C Outline of Proof The solution set which is certainly a subset of C is a vector space because of the superposition principle The dimension comes from compu tations with the Wronskian where it is shown that one needs exactly n linearly independent solutions to nd the general solution to Ly 0 We close with a statement about the general rst order system Proposition 49 Consider the equation 48 X AX where A is a p X p matrix whose entries are continuous in U The solution set is a one dimensional vector space in C1 and the codomain of the solutions is RP SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 MIKE WILLS CONTENTS Introduction Second Order Equations and First Order Systems Vector Spaces Function Spaces FPON E oanH 1 INTRODUCTION The main purpose of this note is to ll in some of the details on the linear algebra side of things for our course For the most part7 you do not need to know this stuff for the class per 567 but perhaps it Will provide a bit of a motivation It is assumed that the reader is familiar With the matrix handouti A second purpose is to clarify some comments made in class 2 SECOND ORDER EQUATIONS AND FIRST ORDER SYSTEMS De nition 21 The most general second order ordinary differential equa tion is one that can be Written in the form 21 gt 7y7y7t 07 With 39i39 nonconstant With respect to Q so that the second derivative term does not just disappearI One can always Write a second order equation as a system of rst order equations by Writing 1 y39 and hence getting the system 39i39 i v y t 0 2 2 7 7 7 t v yr The above formulation is too general for our purposes The usual general formulation of a second order equation is 23 y fy397y7t and the equivalent rst order system is y39 1 2 4 l vfv7y7tA By de ning 2 5 Xy 2 MIKE WILLS and v 26 F lt gt l f l the system can be written in vector form as 27 X F Now F can be thought of as a function of X and t and that motivates the following de nition De nition 22 A rst order system of differential equations is one that can be written in the form 28 X FX t where t can be thought of as a time variable and X is a column vector of one or more time dependent variables7 11 12 In We will assume that there are n equations and so it is natural to refer to the system as n dimensional Remark 23 This is not the most general form7 but it is arguably the most typical Notice that the domain of F is in Rn1 and the codomain is R 39 One can add the extra equation t l and then rewrite the system as Y FY where Y is the column vector X with t adjoined at the end This will make the system autonomous In the case that X consists of only variable7 the system reduces to a single equation which we discussed in chapter 2 Theorem 24 Picard Suppose that F is continuous in a neighbourhood U of mmto Suppose that F11F 2FM are also continuous Then the initial value problem X FX t 29 XOO 1 0 has a unique solution near to Remark 25 The proof of the result starts by de ning 210 X1 X0t FX0sds z and inductively de ning O 211 Xn1 X0 EFXL7 sds to The integration of a vector quantity is understood to be the integration of each term in the vector Note that your correspondent has consistently got this wrong in class The rst term on the right hand side is X0 not Xn The proof works much the same way in any dimension For existence of a solution one merely requires that F be continuous for uniqueness7 Lipschitz continuity suffices Here are some examples of rst order systems in the two dimensional situation SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 3 Example 26 139 yr 212 y y t Example 27 139 7213 mm De nition 28 A rst order system is linear if there exists a matrix At and a column vector bt such that 214 XFXt AXb lf 1 is identically zero7 the system is homogeneous Otherwise7 the system is inhomogeneous As usual7 we will assume that the entries of A and b are continuous Recall that for any square matrix7 B7 00 En 215 expB eB Z W It can be shown that this matrix series always converges The general solution to the homogeneous system is 216 Xht exp t Asdsxo where X0 is an arbitrary vector A particular solution to the inhomogeneous equation can be found using the technique of variation of parameters write 217 Xpa exp t magma Plug Xp into the system and solve for W This will always work By the superposition principle7 the general solution to the inhomogeneous linear system is 218 X XthXp We now introduce some linear algebra that may make the theory discussed in chapter 3 a bit more palatab e 3 VECTOR SPACES We now summarize some of the key points from linear algebra that are discussed in distilled form in our text This is an overview7 and a somewhat sketchy one at that More details can be garnered from onels favorite linear algebra text De nition 31 Let V be a nonempty set Suppose that V is equipped with two operations7 Jr and such that the following hold for any uVW 6 V7 15 E R iiavaveV iii uvwuvw iv There exists 0 E V such that u0 0u u v uvvu 4 MIKE WILLS vi awv Mew amv vii a BV av BV viii aV W av aw ix 1 V V x There exists 7V E V such that V V 0 Then V is a real vector space Elements of V are vectors The operation is vector addition and the operation is scalar multiplication Remark 32 The above axiomatic characterization asserts that V is closed With re spect to vector addition and scalar multiplication that vector addition is associate commutative has an identity and additive inverses exist that scalar multiplication distributes over vector addition and that the multiplying a vector by one does not change the vector A complex vector space satis es the same axioms but in this situation a and B are arbitrary complex numbers More exotic vector spaces also exist but are of limited interest to us at this time In this handout all vector spaces are real unless noted otherwise Example 33 Our calculus textbook goes to great lengths to stress the difference between ndimensional vectors and coordinates in R This distinction is not observed much in linear algebra By identifying points and vectors we see that R and C are vector spaces Example 34 The set an of m X n matrices is a vector space under matrix addition De nition 35 Let V be a vector space Let U C V With U nonempty Suppose that U is closed under vector addition and scalar multiplication that is Whenever uV E U a E R it turns out that u V au 6 U Then U is a subspace of V Example 36 If V is a vector space then 0 and V are subspaces of V Example 37 The other subspaces of R2 are straight lines that pass through the origin Example 38 The other subspaces of R3 are lines and planes through the origin De nition 39 A set of n vectors S V1Vn in V is linearly indepen dent if n 31 Zuzka 0 ak Ofor all k k1 If S is not linearly independent it is said to be linearly dependent Example 310 The set 0 is linearly dependent Example 311 The set ij k C R3 is linearly independent De nition 312 The span of a set S of n vectors in V is the set n 32 W 04ka l 04k 6 k1 If W V then S spans V 1The latter is a complex Vector space SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 5 Proposition 313 The span of S is a subspace of W De nition 314 If S is a set of n linearly independent vectors in V and the span of S is V7 then S is a basis for V and V is an ndimensional vector space If there is no nite subset of V which spans V7 then V is in nite dimensional The vector space7 07 is zero dimensional Example 315 The vector space R is n dimensional The basis vectors form a basis Example 316 Straight lines that pass through the origin in R2 are one dimen sional subspaces of 2 Example 317 Planes that pass through the origin in R3 are two dimensional subspaces of R3 In the calculus sequence7 we do not deal with in nite dimensional vector spaces7 which are a bit tricky to handle However7 such spaces play a crucial role in analysis We will defer some examples until later in this handout De nition 318 Let V and W be vector spaces Let T V A W be a map such that for any u7 v E V and 15 E R the following equation holds 33 Tau v aTu BTW Then T is a linear operator or linear map from V to W Notation 319 One often writes Tu rather than Tu when T is a linear operator Example 320 If A is an m X n matrix7 then the map 11 gt gt A11 is a linear map from R to Rm As it happens7 these are the only linear operators that every one in math 2280 is likely to have seen at the start of the semester ln particular7 the only linear maps from R to R are those of the form y m1 De nition 321 If there exists a bijective linear map7 T7 between two vector spaces V and W then V and W are isomorphic and T is a isomorphism Example 322 The vector space7 V is isomorphic to itself under the identity map v gt gt v Example 323 One dimensional subspaces of R are isomorphic to each other and to R Example 324 If V and W are isomorphic and U and V are isomorphic7 then U and W are isomorphic Hence7 the relation is isomorphic to7 is an equivalence relation Example 325 If V and W are both ndimensional vector spaces7 then V and W are isomorphic Thus7 up to isomorphism7 there is only one n dimensional vector space Similar assertions can be made for vector spaces of any cardinality but that requires rather advanced set theory in particular7 the axiom of choice 6 MIKE WILLS 4 FUNCTION SPACES Many of the most useful vector spaces are what are called function spaces indeed the ones that we use in this course are exactly that What we mean y a function space is a set whose elements are functions equipped with suitable vector space operations Example 41 We de ne lP n to be the set of all polynomials with real coef cients in one variable of degree less than or equal to n together with the zero polynomial whose degree is unde ned or 700 depending on ones point of view For example 41 P1 mm a0 l a1a0 E R Then lP n is a vector space with p q 2 101 41 and ap 0471 A basis for P1 is the set 11 A basis for P4 is 141312zl Hence lP n is isomorphic to RWA The set of all polynomials is also a vector space this time of in nite dimension and is denoted lP Notice that lP O C P1 C P2 H C P Recalling that sequences are functions whose domain is N any vector space whose vectors are sequences can be thought of as a function space Example 42 The set S of sequences of real numbers is a vector space Given a anb 127 E S we de ne ab an bn and for a E R aa aan lmportant subspaces of S are the socalled little el p spaces For example 42 1aESlZlanl lt00 is the set of all absolutely summable sequences Also 43 lma68llanlltM is the set of all bounded sequences Here M depends on the sequence a but not on n Thus different sequences in loo have different M 7s For p 2 l 44 17 a e s i 2 law lt 00 Notice that 1 C loo for any p 2 l The set lg is notable since it has an inner product which is a generalization of the dot product from vector calculus Hence lg has a notion of orthogonality Example 43 Let U C Rm be a closed set De ne f y U A R17 l f is a function Then F is a vector space with y yt zt and ay t ayt for every y z E f t E U and a R We will primarily be concerned with subspaces of 7 When considering ODEs p will be equal to one When considering systems p will be the number of equations in the system Notice that if U R then P C 7 Probably the most important subspaces of f are the socalled big el p spaces LP To de ne those properly requires a bit of measure theory in particular Lebesgue integration which would be a bit of a distraction We therefore turn to the vector spaces of interest to us in ODEs SUPPLEMENTAL NOTES FOR CHAPTERS 3 AND 4 7 De nition 44 For n E N U 0 we de ne C U as follows 4 5 C U y E f l all nth order partial derivatives of y exist and are continuous We recall that the 0th derivative of a function is just the function itself Furt er7 46 CooU y E f l all partial derivatives of all orders of y exist and are continuous Notation 45 If the set U is immaterial or understood7 we will write C for C U We assume for the remainder of this handout that U C R is an interva Proposition 46 The sets C and C00 are in nitedimensional vector spaces Proof The reader should be able to verify this D Proposition 47 The map L C A 7 given by 47 Lltygt 2am k0 where ak U A R is an arbitrary function is a linear operator We will assume that each ak is continuous We will also assume that an is not the zero function Proposition 48 The set of solutions to the equation Ly 0 is an ndimensional subspace of C Outline of Proof The solution set which is certainly a subset of C is a vector space because of the superposition principle The dimension comes from compu tations with the Wronskian where it is shown that one needs exactly n linearly independent solutions to nd the general solution to Ly 0 We close with a statement about the general rst order system Proposition 49 Consider the equation 48 X AX where A is a p X p matrix whose entries are continuous in U The solution set is a one dimensional vector space in C1 and the codomain of the solutions is RP

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