A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS (ODE's) CHAPTER 5 Theory ad Solutio Usig Matrix Techiques 1. Fudametal Theory of Systems of ODE s 2. Matrix Techique for Solvig Systems with Costat Coefficiets 3. Real Distict Roots 4. Complex Roots 5. Repeated Roots Ch. 5 Pg. 1
Hadout #1 FUNDAMENTAL THEORY OF SYSTEMS OF ODE S Prof. Moseley The geeral form for a first order liear system of ODE s is give by We choose t as the idepedet variable so that we ca use x s as the compoets of the vector-valued fuctio: (We use the traspose otatio to save space.) We make the usual assumptio that p ij(t) ad g i(t) are cotiuous t I = (a,b). This system ca be rewritte usig (the much more compact) matrix otatio as: (3) where P is the square matrix-valued fuctio ad g(t) is the vector-valued forcig fuctio The usual form for a ohomogeeous equatio requires that we rewrite this as Ch. 5 Pg. 2
. (6 which we the rewrite as where t I=(a,b). (8) Recall that R is the set of all real valued matrices. Now let R (I) = {P=( p ij(t) ):I R ) ad be the set of time varyig matrices o the ope iterval I=(a,b) ad m m C(R (I)) = {P=( p ij(t) ):I R :pij C(I) for all i ad j} deotes the set of all elemets i m R (I) where all etries are cotiuous o I. Similarly, A(R (I)) = {P=( p ij(t) ):I R :pij A(I) for all i ad j} deotes the set of all elemets i R (I) where all etries are aalytic o I, C(R (I)) = { = [ x i(t) ]:I R :xi C(I) }, 1 C (R (I)) = {x = [ x i(t) ]:I R :xi (t) exists for all i ad is i C(I) }, ad A(R (I)) = { = [ x i(t) ]: I R :x i(t) A(I) }. Now assume P(t) C(R (I)) ad g(t) C(R (I)). 1 The L is a operator that maps vector-valued fuctios i C (R (I)) to vector-valued fuctios i C(R (I)). To solve (6) meas to fid all that map to. Sice algebraically, we ca treat a collectio of vector-valued fuctios (with the appropriate defiitios of vector additio ad scalar multiplicatio) as a vector space, we ca view o this operator as mappig 1 oe vector space ito aother. I this case we are mappig C (R (I); that is, the set of vectorvalued fuctios which have a derivatives that is cotiuous o the iterval of validity I = (a,b) 1 ito the space C (R (I)) of cotiuous vector-valued fuctio o I. If P(t) A(R (I)), the L maps A(R (I)) to A(R (I)). We ow review ad apply the liear theory previously developed ad applied to secod order liear ODE s ad higher order liear ODE s to first order liear systems of ODE s, that is to the operator L give i (8) above. We begi by reviewig the defiitio of a liear operator. A fuctio or mappig T from oe vector space V to aother vector space W is ofte called a operator. We write T:V W. If we wish to thik geometrically rather tha algebraically we might call T a trasformatio. DEFINITION #1. A operator T:V W is said to be liear if x-y V ad scalars á,â we have (9) Recall that we ca divide our check that T is a liear operator ito two steps by usig the followig theorem. THEOREM #1. A operator T is liear if ad oly if the followig two properties hold: Ch. 5 Pg. 3
i) ii) á a scalar ad 1 THEOREM #2. Assume that P(t) C (R (I)) where I = (a,b). The L defied by (8) is a liear 1 1 operator from C (R (I)) to C(R (I)) ad the ull space of L is a subset of C (R (I)). If P(t) A(R (I)), the L defied by (8) is a liear operator from A(R (I)) to A(R (I)) ad N(L) A(R (I)). I either case, dim N(L) =. Hece the solutio of the homogeeous equatio x I = (a,b) = iterval of validity (12) has the form where is a basis for N(L) ad c i = 1,, are arbitrary costats. i, Sice we kow that the dimesio of the ull space N(L) is, if we have a set of solutios to the homogeeous equatio (1), to show that it is a basis of the ull space N(L), it is 1 sufficiet to show that it is a liearly idepedet set. As applied to C (R (I)), C(R (I)), ad A(R (I)), the defiitio of liear idepedece is as follows: 1 DEFINITION #2. The set S = {x 1, x 2, x 3,..., x } C (R (I)) is said to be liearly idepedet o I = (a,b) if the oly solutio to c,c,c,, c R such that 1 2 3 c 1 + c 2 + + c = 0 t R (14) is the trivial solutio, c 1 = c 2 = c = 0. Otherwise S is liearly depedet o the iterval I. The followig theorem is i some sese just a restatemet of the defiitio. 1 THEOREM #3. The set S = {x 1, x 2, x 3,..., x } C (R (I)) is liearly idepedet o I = (a,b) if ad oly if oe vector -valued fuctio ca be writte as a liear combiatio of the other vector -valued fuctios. PROCEDURE. To show that is liear idepedet it is stadard to assume (8) ad try to show c 1 = c 2 = c 3 = = c = 0. Ch. 5 Pg. 4
DEFINITION #3. If where I = (a,b) ad the is called the Wroski determiat or the Wroskia of at the poit t. THEOREM #4. The ull space N(L) of the operator defied by L i (1) above has dimesio. Hece the solutio of the homogeeous equatio has the form x I = (a,b) = iterval of validity (16) where is a basis for N(L) ad c, i = 1,, are arbitrary costats. Give a set of i solutios to L[y] = 0, to show that they are liearly idepedet solutios, it is sufficiet to compute the Wroskia. where i = 1,, ad show that it is ot equal to zero o the iterval validity. THEOREM #5. The ohomogeeous equatio Ch. 5 Pg. 5
t I = (a,b) = iterval of validity (19) has at least oe solutio if the fuctio g is cotaied i the rage space of L, R(L). If this is the case the the geeral solutio of (19) is of the form (20) where is a particular (i.e. ay specific) solutio to (19) ad y is the geeral (e.g. a formula h for all) solutios of (16). Sice N(L) is fiite dimesioal with dimesio we have where is a basis of the ull space N(L). EXERCISES o Fudametal Theory of Systems of ODE S EXERCISE #1 (a) Compute if the operator L is defied by w (b) Compute L[ö] if the operator L is defied by L[x] = x - P(t)x where EXERCISE #2. Directly usig the Defiitio (DUD) or by usig Theorem 1, prove that the 1 followig operators L[y] which map the vector space C (R (I)) (the set of vector-valued fuctio which have a derivative that is cotiuous o the iterval of validity I) ito the space C(R (I)) of cotiuous vector-valued fuctios o I are liear. (a) where Ch. 5 Pg. 6
(b) where EXERCISE #3. Determie (ad prove your aswer directly usig the defiitio (DUD), if the 1 followig sets are.i. or.d i C (R (I); t 2 T 3t 2 T t 2 T t 2 T (a) S = { [ e, si t, 3t ] [e, si t, 3t ] } (b) S = { [3e, 3 si t, 6t ] [e, si t, 3t ] } Hit: Sice (14) must hold t R, as your first try, pick several (distict) values of t to show (if possible) that c 1 = c 2 = c 3 = = c = 0. If (14) must hold t R the it must hold for ay particular value of t. If this is ot possible fid c,c,c,, c ot all zero such that (14) holds 1 2 3 t R. Exhibitig (8) with these values provide coclusive evidece that is liearly depedet. EXERCISE #4. Compute the Wroskia W(x 1,, x ;t) of the followig: (a) t 2 T 3t 2 T t -t [e, si t, 3t ],[e, si t, 3t ],[e,e, si t] (b) at bt [ si t, cos t, t], [e, e, 0], [si t, 0, 0] Ch. 5 Pg. 7
Hadout #2 MATRIX TECHNIQUE FOR SOLVING SYSTEMS Prof. Moseley WITH CONSTANT COEFFICIENTS Recall the homogeeous equatio: (1) where L is a liear operator of the form We cosider the special case where P is a costat matrix. For coveiece of otatio, we cosider: where (3) Note that sice the coefficiet matrix is costat, it is cotiuous for all t R ad the iterval of validity is the etire real lie. Also L maps A(R (I) back to A(R (I). Applyig a previous theorem we obtai: THEOREM. Let be a set of solutios to the homogeeous equatio (3). The the followig are equivalet (i.e. they happe at the same time). a. The set S is liearly idepedet. (This is sufficiet for S to be a basis of the ull space N(L) of the liear operator L[y] sice the dimesio of N(L) is.) b. c. All solutios of (3) ca be writte i the form Ch. 5 Pg. 8
where c i,i=1,, are arbitrary costats. That is, sice S is a basis of N(L) it is a spaig set for N(L) ad hece every vector-valued fuctio i N(L) ca be writte as a liear combiatio of the vector-valued fuctios i S. We ote that by this theorem we have reduced the problem of fidig the geeral solutio of the homogeeous equatio (3) to the fidig of liearly idepedet fuctios We ow develop a techique for solvig first order liear homogeeous systems with costat coefficiets (I.E., P(t) = A is a costat matrix) by fidig the vector-valued fuctios We guess that there may be solutios to (3) of the form (5) where r is a costat ad We attempt to determie r ad is a costat vector, both to be determied later. by substitutig x ito the ODE ad obtaiig a coditio o r ad i order that (3) have a solutio of the form (5). Usig our stadard techique for substitutig ito a liear equatio we obtai: We eed to first check that ideed computig the derivative yields PROOF. STATEMENT REASON defiitio of otatio = defiitio of Ch. 5 Pg. 9
= defiitio of = defiitio of scalar mult = defiitio of derivative of a matrix. = Theorems from calculus = defiitio of scalar multiplicatio defiitio of Next we show that (6) PROOF. STATEMENT REASON Defiitio of Defiitio of scalar multiplicatio. Ch. 5 Pg. 10
Defiitio of A Def of matrix mult Scalar arithmetic. Ch. 5 Pg. 11
Scalar arithmetic. Defiitio of matrix multiplicatio. Defiitio of A ad Hece (6) becomes rt Sice e 0, (it appears i each compoet of the vector equatio o both sides of the equatio), we obtai the eige value problem: Eige Value Problem (8) Ch. 5 Pg. 12
Hece we have chaged the calculus problem of solvig a first order system of ODE s with costat coefficiets to a eige value problem for a matrix. (Hece o ati derivatives eed be computed.) Recall that to solve the eige value problem for a matrix, we must fid the zeros (i.e. th the eige values) of the degree polyomial p(r) = det (A - ri). Thus we obtai the auxiliary equatio: p(r) = det (A - ri) = 0 Auxiliary Equatio (9) I accordace with the fudametal theorem of algebra, (9) has solutios i.e. the left had side (LHS) ca be factored ito (r -r 1)(r - r 2) (r - r ) = 0. (10) Hece your factorig skills for higher degree polyomial equatios are a must. Two or more of the factors i Equatio (8) may be the same. This makes the sematics of the stadard verbiage sometimes cofusig, but oce you uderstad what is meat, there should be o problem. As i secod order equatios, we speak of the special case of repeated roots.) Ch. 5 Pg. 13
Hadout #3 DISTINCT ROOTS Prof. Moseley We illustrate the procedure with a example. EXAMPLE. Solve where SOLUTION. Lettig Step 1. Solve the eige value problem. we obtai the eige value problem Step 1a. Fid the eige values. p(r) = det (A - ri) p(r) = 0 r 1 = 2, r 2 = -3 Step 1b. Fid the associated eige vector. (Let r = 2 Solve reduce A - 2I R 2 - (-4)R 1 Hece the ifiite family of solutios (the vector i the ull space of A - 2I) is give by: r = -3 Solve reduce A + 3I = R 2 - (1)R1 Hece the ifiite family of solutios (the vector i the ull space of A - 2I) is give by: Ch. 5 Pg. 14
TABLE Eige Values Associated Eige Vectors (Basis of the ull space) r 1 = 2 r 2 = -3 Step 2. Write the solutio to the system of ODEs. Let Hece, i vector form we have the solutio as or i scalar form as Ch. 5 Pg. 15