FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems

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1 FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from ha in h book, I fl ha nos ha closly follow h lcur prsnaion migh b apprciad Conns Inroducion o Firs-Ordr Sysms Normal Form and Soluions Iniial-Valu Problms Rcasing Highr-Ordr Problms as Firs-Ordr Sysms 4 4 Numrical Mhods 5 5 Applicaion: Tank Problms 7 Linar Sysms: Gnral Mhods and Thory Iniial-Valu Problms 9 Homognous Sysms 0 Wronskians and Fundamnal Marics 4 Naural Fundamnal Marics 5 5 Nonhomognous Sysms and Grn Marics (no covrd) 7 B Appndix: Vcors and Marics B Vcor and Marix Opraions 0 B Invribiliy and Invrss 4

2 Inroducion o Firs-Ordr Sysms Normal Forms and Soluions For h rmaindr of h cours w will sudy firs-ordr sysms of n ordinary diffrnial quaions for funcions x j (), j,,, n ha can b pu ino h normal form () dx d f (, x, x,, x n ), dx d f (, x, x,, x n ), dx n f n (, x, x,, x n ) d W say ha n is h dimnsion of his sysm Sysm () can b xprssd mor compacly in vcor noaion as dx () d f(,x), whr x and f(,x) ar givn by h n-dimnsional column vcors x f (, x, x,, x n ) f (, x, x,, x n ) x x x n, f(,x) f n (, x, x,, x n ) W hrby xprss h sysm of n quaions () as h singl vcor quaion () W say x, x,, x n ar h nris of h vcor x Similarly, w say ha h funcions f (, x, x,, x n ), f (, x, x,, x n ),, f n (, x, x,, x n ) ar h nris of h vcor-valud funcion f(,x) Rmark W will us boldfac, lowrcas lrs lik x and f o dno column vcors Ohr common noaions includ an undrlin lik x and f, or an arrow lik x and f Som advancd books do no us any spcial noaion for vcors, bu xpc h radr o rcall wha ach lr rprsns from whn i was inroducd You should rcall from muli-variabl Calculus wha i mans for a vcor-valud funcion u() o b ihr coninuous or diffrniabl a a poin W say u() is coninuous a im if vry nry of u() is coninuous a W say u() is diffrniabl a im if vry nry of u() is diffrniabl a Givn hs dfiniions, w dfin wha i mans for a vcor-valud funcion u() o b ihr coninuous, diffrniabl, or coninuously diffrniabl ovr a im inrval W say u() is coninuous ovr a im inrval ( L, R ) if i is coninuous a vry in ( L, R ) W say u() is diffrniabl ovr a im inrval ( L, R ) if i is diffrniabl a vry in ( L, R ) W say u() is coninuously diffrniabl ovr a im inrval ( L, R ) if i is diffrniabl ovr ( L, R ) and is drivaiv is coninuous ovr ( L, R )

3 W ar now rady o dfin wha w man by a soluion of sysm () Dfiniion W say ha x() is a soluion of sysm () ovr a im inrval ( L, R ) whn x() is diffrniabl a vry in ( L, R ); f(,x()) is dfind for vry in ( L, R ); quaion () holds a vry in ( L, R ) Rmark This dfiniion is similar o dfiniions of soluions o singl diffrnial quaions ha w gav arlir Th firs poin sas ha h righ-hand sid of h quaion maks sns Th scond poin sas ha h lf-hand sid of h quaion maks sns Th hird poin sas ha h wo sids ar qual Iniial-Valu Problms W will considr iniial-valu problms of h form () dx d f(,x), x( I) x I Hr is h iniial im, x I is h iniial valu or iniial daa, and x( I ) x I is h iniial condiion Blow w will giv condiions on f(,x) ha insur his problm has a uniqu soluion ha xiss ovr som im inrval ha conains I W bgin wih a dfiniion Dfiniion L S b a s in R R n A poin ( o,x o ) is said o b in h inrior of S if hr xiss a box ( L, R ) (x L, xr ) (xl n, xr n ) ha conains h poin ( o,x o ) and also lis wihin h s S Our basic xisnc and uniqunss horm is h following Thorm L f(,x) b a vcor-valud funcion dfind ovr a s S in R R n such ha f is coninuous ovr S, f is diffrniabl wih rspc o ach x i ovr S, ach xi f is coninuous ovr S Thn for vry inial im I and vry iniial valu x I such ha ( I,x I ) is in h inrior of S hr xiss a uniqu soluion x() o iniial-valu problm () ha is dfind ovr som im inrval (a, b) such ha I is in (a, b), {(,x()) : (a, b)} lis wihin h inrior of S Morovr, x() xnds o h largs such im inrval and x () is coninuous ovr ha im inrval Rmark This is no h mos gnral horm w could sa, bu i applis asily o h firs-ordr you will fac in his cours I assrs ha h iniial-valu problm () has a uniqu soluion x() ha will xis unil (,x()) lavs h inrior of S

4 4 Rcasing Highr-Ordr Problms as Firs-Ordr Sysms Many highrordr diffrnial quaion problms can b rcas in rms of a firs-ordr sysm in h normal form () For xampl, vry n h -ordr ordinary diffrnial quaion in h normal form y (n) g (, y, y,, y (n )), can b xprssd as an n-dimnsional firs-ordr sysm in h form () wih x y y dx d f(,x) x n g(, x, x,, x n ), whr x x x x n y (n ) Noic ha h firs-ordr sysm in xprssd solly in rms of h nris of x Th dicionary ha rlas x o y, y,, y (n ) is givn as a spara quaion Exampl Rcas as a firs-ordr sysm y + yy + y cos() Soluion Bcaus his singl quaion is hird ordr, h firs-ordr sysm will hav dimnsion hr I will b d x x x x, whr x x y y d x cos() x x x x y Mor gnrally, vry d-dimnsional m h -ordr ordinary diffrnial sysm in h normal form y (m) g (,y,y,,y (n )), dx d f(,x) x m g(,x,x,,x m ), whr x x x x m can b xprssd as an md-dimnsional firs-ordr sysm in h form () wih x y y y (m ) Hr ach x k is a d-dimnsional vcor whil x is h md-dimnsional vcor consrucd by sacking h vcors x hrough x m on op of ach ohr Exampl Rcas as a firs-ordr sysm x x x x 4 q + f (q, q ) 0, q + f (q, q ) 0 Soluion Bcaus his wo dimnsional sysm is scond ordr, h firs-ordr sysm will hav dimnsion four I will b x x q d d x 4 f (x, x ), whr x x q q f (x, x ) x 4 q

5 Whn facd wih a highr-ordr iniial-valu problm, w us h dicionary o obain h iniial valus for h firs-ordr sysm from hos for h highr-ordr problm Exampl Rcas as an iniial-valu problm for a firs-ordr sysm y y 0, y(0), y (0), y (0) 5, y (0) 4 Soluion Th firs-ordr iniial-valu problm is x x x (0) d x d x x x 4, x (0) x (0) 5, x 4 (0) 4 x 4 x whr x x x x 4 y y, Rmark W can also find singl highr-ordr quaions ha ar saisfid by h nris of a firs-ordr sysm W will no discuss how his is don bcaus i is no as usful 4 Numrical Mhods On advanag of xprssing an iniial-valu problm in h form of a firs-ordr sysm is ha w can hn apply all h numrical mhods ha w sudid arlir in h sing of singl firs-ordr quaions In fac, h mos common way in which numrical mhods ar applid o consruc a numrical approximaion of h soluion o an iniial-valu problm for a highr-ordr quaion is o rcas i as an iniial-valu problm for a firs-ordr sysm and hn apply such numrical mhods Suppos w wish o consruc a numrical approximaion ovr h im inrval [ I, F ] o h soluion x() of h iniial-valu problm dx (4) d f(,x), x( I) x I A numrical mhod slcs ims { n } N n0 such ha I 0 < < < < N < N F, and compus vcors {x n } N n0 such ha x 0 x I and x n approximas x( n ) for n,,, N If w do his by using N uniform im sps (as w did arlir) hn w s h F I N, and n I + nh for n 0,,, N, whr h is calld h im sp Blow w show ha h vcors {x n } N n0 can b compud asily by h four xplici mhods ha w sudid arlir for scalar-valud quaions: h xplici Eulr, Rung-radzoidal, Rung-midpoin, and Rung-Kua mhods Th only modificaion ha w nd o mak in hs xplici mhods is o rplac h scalar-valud dpndn variabls and funcions wih vcor-valud ons Th jusificaions of hs mhods also carry ovr upon making h sam modificaion Rmark Implici mhods such as h implici Eulr mhod ar ofn xrmly usful for compuing approxima soluions for firs-ordr sysms, bu ar mor complicad o implmn bcaus hy rquir h numrical soluion of algbraic sysms, which is byond h scop of his cours y y 5

6 6 Explici Eulr Mhod Th vcor-valud vrsion of his mhod is as follows S x 0 x I and hn for n 0,, N cycl hrough f n f( n,x n ), x n+ x n + hf n, whr n I + nh Rmark Lik is scalar-valud vrsion, his mhod is firs-ordr Rung-Trapzoidal Mhod Th vcor-valud vrsion of his mhod is as follows S x 0 x I and hn for n 0,, N cycl hrough f n f( n,x n ), fn+ f( n+, x n+ ), x n+ x n + hf n, x n+ x n + h[f n + f n+ ], whr n I + nh Rmark Lik is scalar-valud vrsion, his mhod is scond-ordr I rquirs wic as many funcion valuaions pr im sp as h xplici Eulr mhod Bcaus i is scond ordr, his mhod ofn ouprforms h xplici Eulr mhod bcaus h sam rror ofn can b ralizd wih a im sp ha is mor han wic as larg Rung-Midpoin Mhod Th vcor-valud vrsion of his mhod is as follows S x 0 x I and hn for n 0,, N cycl hrough f n+ f n f( n,x n ), f( n+,x n+ ), x n+ x n + hf n, x n+ x n + hf n+ whr n I + nh and n+ I + (n + )h Rmark Lik is scalar-valud vrsion, his mhod is scond-ordr I has h sam numbr of funcion valuaions pr im sp as h Rung-rapziodal mhod Bcaus hy ar h sam ordr, hir prformancs ar comparabl Rung-Kua Mhod Th vcor-valud vrsion of his mhod is as follows S x 0 x I and hn for n 0,, N cycl hrough fn+ f n+ f n f( n,x n ), f( n+ f( n+, x n+ ), ),,x n+ fn+ f( n+, x n+ ), whr n I + nh and n+ x n+ x n+ x n + hf n, x n + h f n+,, x n+ x n + hf n+ x n+ x n + h[ f 6 n + f n+ + f n+ + f ] n+, I + (n + )h Rmark Lik is scalar-valud vrsion, his mhod is fourh-ordr I rquirs wic as many funcion valuaions pr im sp as ihr scond-ordr mhod and four ims mor han h xplici Eulr mhod Howvr, bcaus i is fourh ordr, h sam rror ofn can b ralizd wih a im sp ha is mor han wic as larg as ha for ihr scond-ordr mhod In such cass, his mhod ouprforms all h forgoing mhods Varians of his mhod ar among h mos widly usd numrical mhods for approximaing h soluion of iniial-valu problms Th varian usd by h MATLAB command od45 is h Dormand-Princ mhod,

7 Rmark In addiion o h four mhods givn abov, hr ar vcor-valud vrsions of h classical hird-ordr mhods du o Hun and Kua Hun Mhod S x 0 x I and hn for n 0,, N cycl hrough f n+ f n+ f n f( n,x n ), f( n+ f( n+,x n+ ), ),,x n+ x n+ x n+ x n + hf n, x n + hf n+, x n+ x n + 4 h[ f n + f n+ whr n I + nh, n+ I + (n + )h, and n+ I + (n + )h Kua Mhod S x 0 x I and hn for n 0,, N cycl hrough f n+ f n f( n,x n ), f( n+,x n+ ), fn+ f( n+, x n+ ), whr n I + nh and n+ x n+ I + (n + )h x n + hf n, ], ], x n+ x n + h [ f n + f n+ x n+ x n + h[ f 6 n + 4f n+ + f ] n+, 5 Applicaion: Tank Problms Firs-ordr sysms aris in many applicaions In his scion w show hy aris from problms of dscribing inrconncd anks Ths rprsn a broad class of problms ha dscrib h ranspor of som quaniy ino and ou of anks or ohr volums Th quaniy migh b a fluid lik war, oil, or air, or i migh b a subsanc lik a solu or poluan ha is carrid along by a fluid Th anks migh b wll-dfind volums lik ponds, laks, or rooms in a building Ths problms li a h har of many numrical simulaions of fluids In such problms w consruc an iniial-valu problm saisfid by h amouns Q i of som quaniy in h ank i Th associad sysm of ordinary diffrnial quaion will consis of quaions in h form dq i RATE IN i RATE OUT i, d whr RATE IN i is h ra h quaniy nrs ank i whil RATE OUT i is h ra h quaniy xis h ank i Thr will b on such quaion for ach ank For som problms RATE IN i and RATE OUT i will b givn xplicily in h problm A ohr ims hy will b givn in rms of ohr varibls in h problm Th way in which his is don is similar o h way w did i for h ank problms w sudid arlir ha involvd jus on ank Exampl Considr wo inrconncd anks filld wih brin (sal war) Th firs ank conains 4 lirs and h scond conains 5 lirs Brin wih a sal concnraion of 9 grams pr lir flows ino h firs ank a 5 lirs pr hour Wll-sirrd brin flows from h firs ank ino h scond a 7 lirs pr hour, from h scond ino h firs a lirs pr hour, from h firs ino a drain a lir pr hour, and from h scond ino a drain a 4 lirs pr hour A 0 hr ar 76 grams of sal in h firs ank and grams in h scond Giv an iniial-valu problm ha govrns h amoun of sal in ach ank as a funcion of im 7

8 8 Soluion L V () and V () b h volums (li) of brin in h firs and scond ank a im minus L S () and S () b h mass (gr) of sal in h firs and scond ank a im minus Bcaus mixurs ar assumd o b wll-sirrd, h sal concnraion of h brin in h anks a im ar C () S ()/V () and C () S ()/V () rspcivly In paricular, hs will b h concnraions of h brin ha flows ou of h rspciv ank W hav h following picur 9 gr/li 5 li/hr C () gr/li li/hr V () li S () gr V (0) 4 li S (0) 76 gr C () gr/li 7 li/hr C () gr/li li/hr V () li S () gr V (0) 5 li S (0) gr C () gr/li 4 li/hr W ar askd o wri down an iniial-valu problm ha govrns S () and S () Th ras work ou so hr will always b V () 4 lirs of brin in h firs ank and V () 5 lirs in h scond Thn S () and S () ar govrnd by h iniial-valu problm ds S d 5 S 4 7 S 4, S (0) 76, ds S d 4 7 S 5 S 5 4, S (0) This can b simplifid o ds d 45 + S 5 4 S, S (0) 76, ds d S 6 7 S 5, S (0) Rmark Th sysm of ordinary diffrnial quaions drivd in h prvious xampl is linar, which is h yp of sysm w will sudy nx

9 9 Linar Sysms: Gnral Mhods and Thory Th n-dimnsional firs-ordr sysm () is calld linar whn i has h form () dx d a ()x + a ()x + + a n ()x n + f (), dx d a ()x + a ()x + + a n ()x n + f (), dx n a n ()x + a n ()x + + a nn ()x n + f n () d Th funcions a jk () ar calld cofficins whil h funcions f j () ar calld forcings W can us marix noaion o compacly wri linar sysm () as dx () A()x + f(), d whr x and f() ar h n-dimnsional column vcors x f () x x, f() f (), x n f n () whil A() is h n n marix a () a () a A() () a () a n () a n () a n () a n () a nn () W call A() h cofficin marix and f() h forcing vcor Sysm () is said o b homognous if f() 0 and nonhomognous ohrwis Th produc A()x apparing in sysm () dnos column vcor ha rsuls from h marix muliplicaion of h marix A() wih h column vcor x Th sum apparing in () dnos column vcor ha rsuls from h marix addiion of h column vcor A()x wih h column vcor f() Ths marix opraions ar prsnd in Appndix B Iniial-Valu Problms W will considr linar iniial-valu problms in h form dx () d A()x + f(), x( I) x I, whr x I is calld h vcor of iniial valus, or simply h iniial vcor A major hm of his scion is ha for vry fac ha w sudid abou highrordr linar quaions hr is an analogous fac abou linar firs-ordr sysms For xampl, h basic xisnc and uniqunss horm is h following

10 0 Thorm If A() and f() ar coninuous ovr h im inrval ( L, R ) hn for vry iniial im I in ( L, R ) and vry iniial vcor x I h iniial-valu problm () has a uniqu soluion x() ha is coninuously diffrniabl ovr ( L, R ) Morovr, if A() and f() ar k-ims coninuously diffrniabl ovr h im inrval ( L, R ) hn x() will b is (k + )-ims coninuously diffrniabl ovr ( L, R ) You should b abl o us h Basic Exisnc and Uniqunss Thorm o idnify h inrval of dfiniion for soluions of h iniial-valu problm () This is don vry much lik h way you idnifid inrvals of dfiniion for soluions of highr-ordr linar quaions Spcifically, if x() is h soluion of h iniial-valu problm () hn is inrval of dfiniion will b ( L, R ) whnvr: vry nry of h cofficin marix A() and h forcing vcor f() ar coninuous ovr ( L, R ), h iniial im I is in ( L, R ), an nry of ihr h cofficin marix or h forcing vcor is undfind a ach of L and R You can do his bcaus h firs wo bulls along wih h Basic Exisnc and Uniqunss Thorm imply ha h inrval of dfiniion will b a las ( L, R ), whil h las wo bulls along wih our dfiniion of soluion imply ha h inrval of dfiniion can b no biggr han ( L, R ) bcaus h quaion braks down a L and R This argumn works whn L or R Homognous Sysms Jus as wih highr-ordr linar quaions, h ky o solving a firs-ordr linar sysm () is undrsanding how o solv is associad homognous sysm (4) dx d A()x W will assum hroughou his scion ha h cofficin marix A() is coninuous ovr an inrval ( L, R ), so ha Throrm can b applid W will xploi h following propry of homognous sysms Thorm (Linar Suprposiion) If x () and x () ar soluions of sysm (4) hn so is c x () + c x () for any valus of h consans c and c Mor gnrally, if x (), x (),, x m () ar m soluions of sysm (4) hn so is h linar combinaion (5) x() c x () + c x () + + c m x m (), for any valus of h consans c, c,, c m Rmark Hr x (), x (),, x m () dno m diffrn vcor-valud soluions of h sysm (4), and should no b confusd wih x (), x (),, x m (), which dno h firs m nris of h vcor-valud funcion x()

11 Rason Bcaus x (), x (), x m () solv (4), a dirc calculaion saring from h linar combinaion (5) shows ha dx d () d ( c x () + c x () + + c m x m () ) d dx dx m d () dx c d () + c d () + + c m c A()x () + c A()x () + + c m A()x m () A() ( c x () + c x () + + c m x m () ) A()x() Thrfor x() givn by h linar combinaion (5) solvs sysm (4) Rmark This horm sas ha any linar combinaion of soluions of (4) is also a soluion of (4) I hrby provids a way o consruc a whol family of soluions from a fini numbr of hm Now considr h iniial-valu problm dx (6) d A()x, x( I) x I Suppos w know n diffrn soluions of (4), x (), x (),, x n () I is naural o ask if w can consruc h soluion of h iniial-valu problm (6) as a linar combinaion of x (), x (),, x n () S (7) x() c x () + c x () + + c n x n () By h suprposiion horm his is a soluion of (4) W only hav o chck ha valus of c, c,, c n can b found so ha x() will also saisfy h iniial condiions in (6) namly, so ha x I x( I ) c x ( I ) + c x ( I ) + + c n x n ( I ) Ψ( I )c, whr c is h n column vcor givn by whil Ψ( I ) is h n n marix givn by c ( c c c n ) T, Ψ( I ) ( x ( I ) x ( I ) x n ( I ) ) This noaion indicas ha h k h column of Ψ( I ) is h column vcor x k ( I ) In his noaion h qusion bcoms whhr hr is a vcor c such ha Ψ( I )c x I, for vry x I This linar algbraic sysm will hav a soluion for vry x I if and only if h marix Ψ( I ) is invribl, in which cas h soluion is uniqu and is givn by c Ψ( I ) x I Of cours, h marix Ψ( I ) is invribl if and only if d ( Ψ( I ) ) 0

12 Wronskians and Fundamnal Marics Givn any s of n soluions x (), x (),, x n () o h homognous quaion (4), w dfin is Wronskian by (8) W[x,x,,x n ]() d ( x () x () x n () ) Th Abl Thorm for firs-ordr sysms is (9) d d W[x,x,,x n ]() r ( A() ) W[x,x,,x n ](), whr r ( A() ) dnos h rac of A(), which is givn by (0) r ( A() ) a () + a () + + a nn () Upon ingraing h firs-ordr linar quaion (9) w s ha ( () W[x,x,,x n ]() W[x,x,,x n ]( I ) xp I r ( A(s) ) ) ds As was h cas for highr-ordr quaions, his shows ha if h Wronkian is nonzro somwhr hn i is nonzro vrywhr, and ha if i is zro somwhr, i is zro vrywhr! Again analogous o h cas for highr-ordr quaions, w hav h following dfiniions Dfiniion A s of n soluions x (), x (),, x n () o h n-dimnsional homognous linar sysm (4) calld fundamnal if is Wronskian is nonzro Thn h family () x() c x () + c x () + + c n x n () is calld a gnral soluion of sysm (4) Howvr, now w inroduc a nw concp for firs-ordr sysms Dfiniion If x (), x (),, x n () is a fundamnal s of soluions o sysm (4) hn h n n marix-valud funcion () Ψ() ( x () x () x n () ) is calld a fundamnal marix for sysm (4) Som basic facs abou fundamnal marics ar as follows Fac L Ψ() b a fundamnal marix for sysm (4) Thn Ψ() saisfis (4) Ψ A()Ψ, d ( Ψ() ) 0 ; A gnral soluion of sysm (4) is x() Ψ()c ;

13 Rason By () w s ha Ψ () ( x () x () x n () ) ( x () x () x n ()) ( A()x () A()x () A()x n () ) A() ( x () x () x n () ) A()Ψ() Also by () w s ha d ( Ψ() ) d ( x () x () x n () ) W[x,x,,x n ]() 0 I should b clar from () ha h gnral soluion givn by () can b xprssd as x() Ψ()c Exampl Th vcor-valud funcions x () 5, x () ar soluions of h diffrnial sysm d x d x x x Consruc a gnral soluion and a fundamnal marix for his sysm Soluion I is asy o chck ha x () and x () ar ach soluions o h diffrnial sysm Bcaus 5 W[x,x ]() d 5 6, w s ha x () and x () compris a fundamnal s of soluions o h sysm Thrfor a fundamnal marix is givn by Ψ() ( x () x () ) 5 5, whil a gnral soluion is givn by ( 5 x() Ψ()c 5 ) ( c c ) c 5 + c c 5 c Alrnaivly, w can consruc a gnral soluion as 5 c x() c x () + c x () c 5 + c 5 + c c 5 c, Exampl Th vcor-valud funcions + x (), x (),

14 4 ar soluions of h diffrnial sysm d x 4 x d x x Consruc a gnral soluion and a fundamnal marix for his sysm Soluion I is asy o chck ha x () and x () ar ach soluions o h diffrnial sysm Bcaus + W[x,x ]() d, w s ha x () and x () compris a fundamnal s of soluions o h sysm Thrfor a fundamnal marix is givn by Ψ() ( x () x () ) +, whil a gnral soluion is givn by + x() Ψ()c c c Alrnaivly, w can consruc a gnral soluion as + x() c x () + c x () c + c c ( + ) + c c + c c ( + ) + c c + c Rmark Th soluions x () and x () wr givn o you in h problms abov Scions and 4 will prsn mhods by which w can consruc a fundamnal s of soluions (and hrfor a fundamnal marix) for any homognous sysm wih a consan cofficin marix For sysms wih a variabl cofficin marix you will always b givn soluions Rmark Any marix-valud funcion Ψ() such ha d ( Ψ() ) 0 ovr som im inrval ( L, R ) is a fundamnal marix for h firs-ordr diffrnial sysm x A()x, whr A() Ψ ()Ψ() This can b sn by muliplying (4) on h lf by Ψ() Exampl Find a firs-ordr diffrnial sysm such ha h vcor-valud funcions + x (), x (), compris a fundamnal s of soluions Soluion S Ψ() ( x () x () ) +

15 Bcaus d ( Ψ() ), w s ha Ψ() is invribl S A() Ψ ()Ψ() Thrfor x () and x () ar a fundamnal s of soluions for h diffrnial sysm d x 4 x d x x 4 Naural Fundamnal Marics Considr h iniial-valu problm (5) x A()x(), x( I ) x I L Ψ() b any fundamnal marix for his sysm Thn a gnral soluion of h sysm is givn by x() Ψ()c By imposing h iniial condiion from (5) w s ha x I x( I ) Ψ( I )c Bcaus d ( Ψ( I ) ) 0, h marix Ψ( I ) is invribl and w can solv for c as c Ψ( I ) x I Hnc, h soluion of h iniial-valu problm is (6) x() Ψ()Ψ( I ) x I Now l Φ() b h marix-valud funcion dfind by (7) Φ() Ψ()Ψ( I ) If w diffrnia Φ() and us h fac ha Ψ() is a fundamnal marix for sysm (5) w s ha Φ () ( Ψ()Ψ( I ) ) Ψ ()Ψ( I ) A()Ψ()Ψ( I ) A()Φ() Morovr, from (7) w s ha Φ( I ) Ψ( I )Ψ( I ) I Thrfor Φ() is h soluion of h marix-valud iniial-valu problm (8) Φ A()Φ, Φ( I ) I This shows hr hings Φ() as a funcion of is a fundamnal marix for sysm (5); Φ() is uniquly drmind by h marix-valud iniial-valu problm (8); Φ() is indpndn of our original choic of fundamnal marix Ψ() ha was usd o consruc i in (7) W call Φ() h naural fundamnal marix associad wih h iniial im I 5

16 6 Jus lik i was asy o xprss h soluion of an iniial-valu problm for a highrordr quaion in rms of is associad naural fundamnal ss of soluions, w xprss h soluion of h iniial-valu problm (5) in rms of is associad naural fundamnal marix as simply (9) x() Φ()x I Givn any fundamnal marix Ψ(), w consruc h naural fundamnal marix associad wih h iniial im I by formula (7) Exampl Consuc h naural fundamnal marix associad wih h iniial im 0 for h sysm d x x d x x Us i o solv h iniial-valu problm wih h iniial condiions x (0) 4 and x (0) Soluion W hav alrady sn ha a fundamnal marix for his sysm is 5 Ψ() 5 By formula (7) h naural fundamnal marix associad wih h iniial im 0 is Φ() Ψ()Ψ(0) Thrfor h soluion of h iniial-valu problm is x() Φ()x I Exampl Consuc h naural fundamnal marix associad wih h iniial im for h sysm d x 4 x d x x Us i o solv h iniial-valu problm wih h iniial condiions x () and x () 0 Soluion W hav alrady sn ha a fundamnal marix for his sysm is + Ψ() By formula (7) h naural fundamnal marix associad wih h iniial im is + Φ() Ψ()Ψ() + + +

17 7 Thrfor h soluion of h iniial-valu problm is ( + x() Φ()x I + + 0) 5 Nonhomognous Sysms and Grn Marics W now considr h nonhomognous firs-ordr linar sysm (0) x A()x + f() If x P () is a paricular soluion of his sysm and Ψ() is a fundamnal marix of h associad homognous sysm hn a gnral soluion of sysm (0) is x() x H () + x P (), whr x H () is h gnral soluion of h associad homognous problm givn by () x H () Ψ()c Rcall ha if w know a fundamnal s of soluions o h associad homognous diffrnial quaion hn w can us ihr Variaions of Paramrs or gnral Grn Funcions o consruc a paricular soluion o a nonhomognous n h -ordr linar quaion in rms of n inrgrals Hr w show ha a similar hing is ru for h nonhomognous firs-ordr linar sysm (0) Spcifically, if w know a fundamnal marix Ψ() for h associad homognous sysm hn w can consruc a paricular soluion o h n dimnsional nonhomognous linar quaion in rms of n inrgrals W bgin wih h analog of h mhod of Variaion of Paramrs for h nonhomognous firs-ordr linar sysm (0) W will assum ha A() and f() ar coninuous ovr an inrval ( L, R ) W also will assum ha w know a fundamnal marix Ψ() of h associad homognous sysm This marix will b coninuously diffrniabl ovr ( L, R ) and saisfy Ψ () A()Ψ(), d ( Ψ() ) 0 Bcaus x H () has h form (), w sk a paricular soluion in h form () x P () Ψ()u(), whr u() is a vcor-valud funcion By diffrniaion w s ha x P() ( Ψ()u() ) Ψ ()u() + Ψ()u () A()Ψ()u() + Ψ()u () A()x P () + Ψ()u () By comparing h righ-hand sid of his quaion wih h righ-hand sid of quaion (0), w s ha x P () will solv (0) if u() saisfis Bcaus Ψ() is invribl, w find ha Ψ()u () f() () u () Ψ() f()

18 8 If u P () is a primiiv of h righ-hand sid abov hn a gnral soluion of his sysm has h form u() c + u P () Whn his soluion is placd ino h form (), w find ha a paricular soluion is givn by (4) x P () Ψ()u P () Now l I b any iniial im in ( L, R ) and considr h iniial-valu problm (5) x A()x + f(), x( I ) x I If w ak h paricular soluion of () givn by hn (4) bcoms u P () (6) x P () Ψ() I Ψ(s) f(s) ds, Th soluion of h iniial-valu problm is hn (7) x() Ψ()Ψ( I ) x I + Ψ() W dfin h Grn marix G(, s) by I Ψ(s) f(s) ds (8) G(, s) Ψ()Ψ(s) Thn w can rcas (7) as (9) x() G(, I )x I + I Ψ(s) f(s) ds I G(, s)f(s) ds In paricular, h paricular soluion of (7) ha saisfis x( I ) 0 is givn by (0) x() I G(, s)f(s) ds Th Grn marix has h propry ha for vry I in ( L, R ) h naural fundamnal marix associad wih I is givn by Φ() G(, I ) Exampl Consuc h Grn marix for h sysm d x x 4 + d x x + Soluion W hav sn ha a fundamnal marix associad wih his sysm is 5 Ψ() 5

19 9 Thn by formula (8) h Grn marix is givn by G(, s) Ψ()Ψ(s) 5 5s s 5 5s s 5 s s 5 6s 5s 5s 5 5s 5s 5 s s 5( s) + s 5( s) s 5( s) s 5( s) + s Exampl Consuc h Grn marix for h sysm d x 4 x d x + x Soluion W hav sn ha a fundamnal marix associad wih his sysm is + Ψ() Thn by formula (8) h Grn marix is givn by + G(, s) Ψ()Ψ(s) + s s s + s + s + s s + s s s s + s s

20 0 B Appndix: Vcors and Marics An m n marix A consiss of a rangular array of nris arrangd in m rows and n columns a a a n a () A a a n a jk a m a m a mn W call a jk h jk-nry of A, m h row-dimnsion of A, n h column-dimnsion of A, and m n h dimnsions of A Th nris of a marix can b drawn from any s, bu in his cours hy will b numbrs Spcial kinds of marics includ: m marics ar calld row vcors; n marics ar calld column vcors; n n marics ar calld squar marics Two m n marics A (a jk ) and B (b jk ) ar said o b qual if a jk b jk for vry j,,, m and k,,, n, in which cas w wri A B W will us 0 o dno any marix or vcor ha has vry nry qual o zro A marix or vcor is said o b nonzro if a las on of is nris is no qual o zro B Vcor and Marix Opraions Marix Addiion Two m n marics A (a jk ) and B (b jk ) can b addd o cra a nw m n marix sum A + B, calld h sum of A and B, dfind by A + B ( a jk + b jk ) If marics A, B, and C hav h sam dimnsions hn marix addiion saisfis A + B B + A commuaiviy, (A + B) + C A + (B + C) A A A A + ( A) ( A) + A 0 associaiviy, addiiv idniy, addiiv invrs Hr h marix A is dfind by A ( a jk ) whn A ( ajk ) W dfin marix subracion by A B A + ( B) Scalar Muliplicaion A numbr α and an m n marix A (a jk ) can b muliplid o cra a nw m n marix αa, calld h mulipl of A by α, dfind by αa ( αa jk ) If marics A and B hav h sam dimnsions hn scalar muliplicaion saisfis α(βa) (αβ)a α(a + B) αa + αb (α + β)a αa + βa A A, A A 0A 0, α0 0 associaiviy, disribuiviy ovr marix addiion, disribuiviy ovr scalar addiion, scalar idniy, scalar muliplicaiv nulliy

21 Marix Muliplicaion An l m marix A and an m n marix B can b muliplid o cra a nw l n marix AB, calld h produc of A and B, dfind by AB ( c ik ), whr cik m a ij b jk j Rmark Noic ha for som marics A and B, dpnding only on hir dimnsions, nihr AB nor BA xis; for ohrs xcaly on of AB or BA xiss; whil for ohrs boh AB and BA xis Exampl For h marics A 5 + i6 i i5 4, B, 6 + i + i i h produc AB xiss wih ( i5) + (5 + i6) ( ) ( 4) + (5 + i6) (6 + i) AB ( i5) + (i) ( ) ( 4) + (i) (6 + i) ( + i) ( i5) + ( i) ( ) ( + i) ( 4) + ( i) (6 + i) i0 + i4 7 i0 + i8, 8 i6 5 i6 whil h produc BA dos no xis Rmark Noic ha if A and B ar n n hn AB and BA boh xis and ar n n, bu in gnral Exampl For h marics w s ha whrby AB BA A AB BA! 9, B, AB, BA, Rmark Th abov xampl also shows ha AB 0 dos no imply ha ihr A 0 or B 0!

22 If α is a numbr and A, B, and C ar marics ha hav h dimnsions indicad on h lf hn marix muliplicaion saisfis A B C l m m n l m m n n k l m m n m n l m l m m n (αa)b α(ab) (AB)C A(BC) A(B + C) AB + AC (A + B)C AC + BC associaiviy, associaiviy, lf disribuiviy, righ disribuiviy An idniy marix is a squar marix I in h form I W will us I o dno any idniy marix; h dimnsions of I will always b clar from h conx Idniy marics hav h propry for any m n marix A IA AI A muliplicaiv idniy Noic ha h firs I abov is m m whil h scond is n n Similarly, zro marics hav h propry for any m n marix A 0A 0, A0 0 muliplicaiv nulliy If h firs 0 is l m hn h scond is l n If h hird 0 is n k hn h fourh is m k Marix Conjuga Th conjuga of h m n marix A givn by () is h m n marix A givn by A ( a jk ) If A A hn ach nry of A is ral and A is calld a ral marix If A A hn ach nry of A is imaginary and A is calld a imaginary marix If α is a numbr and A and B ar marics ha hav h dimnsions indicad on h lf hn marix conjuga saisfis A B m n m n m n l m m n m n (A + B) A + B, (αa) αa, AB AB (no no flip), (A) A

23 Marix Transpos Th ranspos of h m n marix A givn by () is h n m marix A T givn by a a a m a A T a a m a n a n a mn If α is a numbr and A and B ar marics ha hav h dimnsions indicad on h lf hn marix ranspos saisfis A B m n m n m n l m m n m n (A + B) T A T + B T, (αa) T αa T, (AB) T B T A T (no flip), ( A T )T A Hrmiian Transpos Th Hrmiian ranspos of h m n marix A givn by () is h n m marix A A T A T If α is a numbr and A and B ar marics ha hav h dimnsions indicad on h lf hn Hrmiian ranspos saisfis A B m n m n m n l m m n m n (A + B) A + B, (αa) αa, (AB) B A (no flip), ( A ) A Exampls For h marics A 5 + i6 i i5 4, B 6 + i + i i, w hav A 5 i6 i + i5 4, B, 6 i i + i + i + i5 A T, B 5 + i6 i i T, 4 6 i i i5 A, B 5 i6 i + i, i

24 4 B Invribiliy and Invrss An n n marix A is said o b invribl if hr xiss anohr n n marix B such ha AB BA I, in which cas B is said o b an invrs of A Fac A marix can hav a mos on invrs Rason Suppos ha B and C ar invrss of A Thn B BI B(AC) (BA)C IC C If A is invribl hn is uniqu invrs is dnod A Fac If A and B ar invribl n n marics and α 0 hn αa is invribl wih (αa) α A ; AB is invribl wih (AB) B A (noic h flip); A is invribl wih ( A ) A; A is invribl wih ( A ) A ; A T is invribl wih ( A T) ( A ) T ; A is invribl wih ( A ) ( A ) Rason Each of hs facs can b chckd by dirc calculaion For xampl, h scond fac is chckd by (AB) ( B A ) ( (AB)B ) A ( A ( BB )) A (AI)A AA I Fac If A and B ar n n marics and AB is invribl hn boh A and B ar invribl wih A B(AB), B (AB) A Rason Each of hs facs can b chckd by dirc calculaion Fac If A is invribl and AB 0 hn B 0 Rason B IB (A A)B A (AB) A 0 0 Fac No all nonzro squar marics ar invribl Rason Earlir w gav an xampl of wo nonzro marics A and B such ha AB 0 Th prvious fac hn implis ha A is no invribl

25 Drminans and Invribiliy Th invrabiliy of a squar marix is characrizd by is drminan Fac A marix A is invribl if and only if d(a) 0 Rason This fac is provd in Linar Algbra courss W will prov i for h spcial cas of marics in h following xampl Exampl For marics h invrs is asy o compu whn i xiss If a b A, c d hn d(a) ad bc If d(a) ad bc 0 hn A is invribl wih A d b d b d(a) c a ad bc c a This rsul follows from h calculaion a b d b ab cd ab + ba (ad bc) c d c a cd + dc cb + da 0 d(a)i 0 This calculaion also shows ha if d(a) ad bc 0 hn A is no invribl Th following is a vry imporan fac abou drminans Fac If A and B ar n n marics hn d(ab) d(a) d(b) Rason W can prov his fac for marics by dirc calculaion If a b f A, B, c d g h hn ( ) () a b f a + bg af + bh d(ab) d d c d g h c + dg cf + dh (a + bg)(cf + dh) (c + dg)(af + bh) acf + adh + bgcf + bgdh caf cbh dgaf dgbh adh + bcfg bch adfg (ad bc)(h fg) d(a) d(b) This rmarkabl fac is provd for n n marics in Linar Algbra courss Rmark Noic ha if d(ab) 0 hn h abov fac implis ha boh d(a) 0 and d(b) 0 This is consisan wih h fac givn arlir ha if AB is invribl hn boh A and B ar invribl Ohr imporan facs abou drminans includ h following Fac If A is an n n marix hn d ( A ) d(a), d ( A T) d(a), d(a ) d(a) Rason I is asy o prov hs facs for marics Ths facs ar provd for n n marics in Linar Algbra courss Rmark Ths ar consisan wih h facs givn arlir ha if A is invribl hn so ar A, A T, and A 5