Systems of First Order Linear Differential Equations

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1 Sysms of Firs Ordr Linr Diffrnil Equions W will now urn our nion o solving sysms of simulnous homognous firs ordr linr diffrnil quions Th soluions of such sysms rquir much linr lgbr (Mh Bu sinc i is no prrquisi for his cours, w hv o limi ourslvs o h simpls insncs: hos sysms of wo quions nd wo unknowns only Bu firs, w shll hv brif ovrviw nd lrn som noions nd rminology A sysm of n linr firs ordr diffrnil quions in n unknowns (n n n sysm of linr quions hs h gnrl form: n n g n n g n n g (* : : : : : : n n n nn n g n Whr h cofficins ij s, nd g i s r rbirry funcions of If vry rm g i is consn zro, hn h sysm is sid o b homognous Ohrwis, i is nonhomognous sysm if vn on of h g s is nonzro 8, Zchry S Tsng D- -

2 8, Zchry S Tsng D- - Th sysm (* is mos ofn givn in shorhnd form s mri-vcor quion, in h form: A g n n nn n n n n n n g g g g : : : : : : : : : : : : : : A g Whr h mri of cofficins, A, is clld h cofficin mri of h sysm Th vcors,, nd g r n :, n :, g g n g g g : For homognous sysm, g is h zro vcor Hnc i hs h form A

3 Fc: Evry n-h ordr linr quion is quivln o sysm of n firs ordr linr quions (This rlion is no on-o-on Thr r mulipl sysms hus ssocid wih ch linr quion, for n > Empls: (i Th mchnicl vibrion quion m u γ u k u F( is quivln o k m γ m F( m No h h sysm would b homognous (rspcivly, nonhomognous if h originl quion is homognous (rspcivly, nonhomognous (ii y y y y is quivln o 8, Zchry S Tsng D- -

4 This procss cn b sily gnrlizd Givn n n-h ordr linr quion n y (n n y (n n y (n y y y g( Mk h subsiuions: y, y, y,, n y (n, nd n y (n Th firs n quions follow husly Lsly, subsiu h s ino h originl quion o rwri i ino h n-h quion nd obin h sysm of h form: : : : : : : n n n n n n n n n g( n No: Th rvrs is lso ru (mosly * Givn n n n sysm of linr quions, i cn b rwrin ino singl n-h ordr linr quion * Th cpions bing h sysms whos cofficin mrics r digonl mrics Howvr, our Eignvcor mhod will nvrhlss b bl o solv hm wihou ny modificion 8, Zchry S Tsng D- -

5 Erciss D-: onvr ch linr quion ino sysm of firs ordr quions y y y y y 9y cos y ( y πy πy 6y Rwri h sysm you found in ( Ercis, nd (b Ercis, ino mri-vcor quion onvr h hird ordr linr quion blow ino sysm of firs ordr quion using ( h usul subsiuions, nd (b subsiuions in h rvrs ordr: y, y, y Dduc h fc h hr r mulipl wys o rwri ch n-h ordr linr quion ino linr sysm of n quions y 6y y y Answrs D-: 9 cos 6 π π ( (b 9 cos ( (b 6 6 8, Zchry S Tsng D- -

6 A rsh ours in ( Mrics Svrl wks worh of mri lgbr in n hour (Rl, w will only sudy h simpls cs, h of mrics Rviw opics: Wh is mri (pl mrics? A mri is rcngulr rry of objcs (clld nris Thos nris r usully numbrs, bu hy cn lso includ funcions, vcors, or vn ohr mrics Ech nry s posiion is ddrssd by h row nd column (in h ordr whr i is locd For mpl, rprsns h nry posiiond h h row nd h nd column of h mri A Th siz of mri Th siz of mri is spcifid by numbrs [numbr of rows] [numbr of columns] Thrfor, n m n mri is mri h conins m rows nd n columns A mri h hs qul numbr of rows nd columns is clld squr mri A squr mri of siz n n is usully rfrrd o simply s squr mri of siz (or ordr n Noic h if h numbr of rows or columns is, h rsul (rspcivly, n, or n m mri is jus vcor A n mri is clld row vcor, nd n m mri is clld column vcor Thrfor, vcors r rlly jus spcil yps of mrics Hnc, you will probbly noic h similriis bwn mny of h mri oprions dfind blow nd vcor oprions h you migh b fmilir wih 8, Zchry S Tsng D- - 6

7 Two spcil yps of mrics Idniy mrics (squr mrics only Th n n idniy mri is ofn dnod by I n I, I, c Propris (ssum A nd I r of h sm siz: AI IA A I n, ny n vcor Zro mrics mrics h conin ll-zro nris Propris: A A A A A Arihmic oprions of mrics (i Addiion / subrcion c b d ± g f h ± c± g b± f d± h 8, Zchry S Tsng D- - 7

8 (ii Sclr Muliplicion b k kb k, for ny sclr k c d kc kd (iii Mri muliplicion c b d g f h bg c dg f cf bh dh Th mri muliplicion AB is dfind only if hr r s mny rows in B s hr r columns in A For mpl, whn A is m k nd B is k n Th produc mri is going o b of siz m n, nd whos ij-h nry, c ij, is qul o h vcor do produc bwn h i- h row of A nd h j-h column of B Sinc vcors r mrics, w cn lso muliply oghr mri nd vcor, ssuming h bov rsricion on hir sizs is m Th produc of mri nd - nry column vcor is c b d y by c dy No : Two squr mrics of h sm siz cn lwys b muliplid oghr Bcus, obviously, hving h sm numbr of rows nd columns, hy sisfy h siz rquirmn oulind bov No : In gnrl, AB BA Indd, dpnding on h sizs of A nd B, on produc migh no vn b dfind whil h ohr produc is 8, Zchry S Tsng D- - 8

9 Drminn (squr mrics only For mri, is drminn is givn by h formul d c b d d bc No: Th drminn is funcion whos domin is h s of ll squr mrics of crin siz, nd whos rng is h s of ll rl (or compl numbrs 6 Invrs mri (of squr mri Givn n n n squr mri A, if hr iss mri B (ncssrily of h sm siz such h AB BA I n, hn h mri B is clld h invrs mri of A, dnod A Th invrs mri, if i iss, is uniqu for ch A A mri is clld invribl if i hs n invrs mri Thorm: For ny mri A is invrs, if iss, is givn by c b d, A d bc d c b Thorm: A squr mri is invribl if nd only if is drminn is nonzro 8, Zchry S Tsng D- - 9

10 8, Zchry S Tsng D- - Empls: L A nd B (i A B ( ( (ii AB On h ohr hnd: BA (iii d(a (, d(b 8 Sinc nihr is zro, s rsul, hy r boh invribl mrics (iv A / / 6 / 6 / (

11 7 Sysms of linr quions (lso known s linr sysms A sysm of linr (lgbric quions, A b, could hv zro, cly on, or infinily mny soluions (Rcll h ch linr quion hs lin s is grph A soluion of linr sysm is common inrscion poin of ll h quions grphs nd hr r only wys s of lins could inrsc If h vcor b on h righ-hnd sid is h zro vcor, hn h sysm is clld homognous A homognous linr sysm lwys hs soluion, nmly h ll-zro soluion (h is, h origin This soluion is clld h rivil soluion of h sysm Thrfor, homognous linr sysm A could hv ihr cly on soluion, or infinily mny soluions Thr is no ohr possibiliy, sinc i lwys hs, ls, h rivil soluion If such sysm hs n quions nd cly h sm numbr of unknowns, hn h numbr of soluion(s h sysm hs cn b drmind, wihou hving o solv h sysm, by h drminn of is cofficin mri: Thorm: If A is n n n mri, hn h homognous linr sysm A hs cly on soluion (h rivil soluion if nd only if A is invribl (h is, i hs nonzro drminn I will hv infinily mny soluions (h rivil soluion, plus infinily mny nonzro soluions if A is no invribl (quivlnly, hs zro drminn 8, Zchry S Tsng D- -

12 8 Eignvlus nd Eignvcors Givn squr mri A, suppos hr r consn r nd nonzro vcor such h A r, hn r is clld n Eignvlu of A, nd is n Eignvcor of A corrsponding o r Do ignvlus/vcors lwys is for ny givn squr mri? Th nswr is ys How do w find hm, hn? Rwri h bov quion, w g A r Th n sp would b o fcor ou Bu doing so would giv h prssion (A r Noic h i rquirs us o subrc numbr from n n n mri Th s n undfind oprion Hnc, w nd o furhr rfind i by rwriing h rm r r I, nd hn fcoring ou, obining (A r I This is n n n sysm of homognous linr (lgbric quions, whr h cofficin mri is (A r I W r looking for nonzro soluion of his sysm Hnc, by h horm w hv jus sn, h ncssry nd sufficin condiion for h isnc of such nonzro soluion, which will bcom n ignvcor of A, is h h cofficin mri (A r I mus hv zro drminn S is drminn o zro nd wh w g is dgr n polynomil quion in rms of r Th cs of mri is s follow: A r I c b d r r c b d r 8, Zchry S Tsng D- -

13 Is drminn, s o, yilds h quion d r c b d r ( r( d r bc r ( d r ( d bc I is dgr polynomil quion of r, s you cn s This polynomil on h lf is clld h chrcrisic polynomil of h (originl mri A, nd h quion is h chrcrisic quion of A Th roo(s of h chrcrisic polynomil r h ignvlus of A Sinc ny dgr n polynomil lwys hs n roos (rl nd/or compl; no ncssrily disinc, ny n n mri lwys hs ls on, nd up o n diffrn ignvlus Onc w hv found h ignvlu(s of h givn mri, w pu ch spcific ignvlu bck ino h linr sysm (A r I o find h corrsponding ignvcors 8, Zchry S Tsng D- -

14 8, Zchry S Tsng D- - Empls: A A r I r r r Is chrcrisic quion is 6 ( ( 6 ( ( d r r r r r r r r Th ignvlus r, hrfor, r nd 6 N, w will subsiu ch of h ignvlus ino h mri quion (A r I For r, h sysm of linr quions is (A r I (A I Noic h h mri quion rprsns dgnrd sysm of linr quions Boh quions r consn mulipls of h quion Thr is now only quion for h unknowns, hrfor, hr r infinily mny possibl soluions This is lwys h cs whn solving for ignvcors Ncssrily, hr r infinily mny ignvcors corrsponding o ch ignvlu

15 Solving h quion, w g h rlion Hnc, h ignvcors corrsponding o r r ll nonzro mulipls of k Similrly, for r 6, h sysm of quions is 6 (A r I (A 6 I 6 Boh quions in his scond linr sysm r quivln o Is soluions r givn by h rlion Hnc, h ignvcors corrsponding o r 6 r ll nonzro mulipls of k No: Evry nonzro mulipl of n ignvcor is lso n ignvcor 8, Zchry S Tsng D- -

16 Two shor-cus o find ignvlus: If A is digonl or ringulr mri, h is, if i hs h form b d, or d, or c d Thn h ignvlus r jus h min digonl nris, r nd d in ll mpls bov If A is ny mri, hn is chrcrisic quion is d r c b d r r ( d r ( d bc If you r fmilir wih rminology of linr lgbr, h chrcrisic quion cn b mmorizd rhr sily s r Trc(A r d(a No: For ny squr mri A, Trc(A [sum of ll nris on h min digonl (running from op-lf o boom-righ] For mri A, Trc(A d 8, Zchry S Tsng D- - 6

17 A shor-cu o find ignvcors (of mri: Similrly, hr is rick h nbls us o find h ignvcors of ny mri wihou hving o go hrough h whol procss of solving sysms of linr quions This shor-cu is spcilly hndy whn h ignvlus r compl numbrs, sinc i voids h nd o solv h linr quions which will hv compl numbr cofficins (Wrning: This mhod dos no work for ny mri of siz lrgr hn W firs find h ignvlu(s nd hn wri down, for ch ignvlu, h mri (A r I s usul Thn w k ny row of (A r I h is no consisd of nirly zro nris, sy i is h row vcor (α, β W pu minus sign in fron of on of h nris, for mpl, (α, β Thn n ignvcor of h mri A is found by swiching h wo nris in h bov vcor, h is, k (β, α Empl: Prviously, w hv sn A Th chrcrisic quion is r Trc(A r d(a r r 6 (r (r 6, which hs roos r nd 6 For r, h mri (A r I is Tk h firs row, (,, which is non-zro vcor; pu minus sign o h firs nry o g (, ; hn swich h nry, w now hv k (, I is indd n ignvcor, sinc i is nonzro consn mulipl of h vcor w found rlir On vry rr occsions, boh rows of h mri (A r I hv ll zro nris If so, h bov lgorihm will no b bl o find n ignvcor Insd, undr his circumsnc ny non-zro vcor will b n ignvcor 8, Zchry S Tsng D- - 7

18 Erciss D-: L 7 nd D ompu: (i D nd (ii D ompu: (i D nd (ii D ompu: (i d(, (ii d(d, (iii d(d, (iv d(d ompu: (i, (ii D, (iii (D, (iv show (D D Find h ignvlus nd hir corrsponding ignvcors of nd D Answrs D-: (i, (ii 8 (i, (ii 8 (i 8, (ii, (iii 6, (iv 6 /8 /8 / /6 /6 (i, (ii 7 /8 /8, (iii, (iv Th / / quliy is no coincidnc In gnrl, for ny pir of invribl mrics nd D, (D D s s (i r, k ; r, k 7s ; s ny nonzro numbr s s (ii r, k ; r, k s / ; s ny nonzro numbr s 8, Zchry S Tsng D- - 8

19 Soluion of sysms of firs ordr linr quions onsidr sysm of simulnous firs ordr linr quions b c d I hs h lrn mri-vcor rprsnion b c d Or, in shorhnd A, if A is lrdy known from con W know h h bov sysm is quivln o scond ordr homognous linr diffrnil quion As rsul, w know h h gnrl soluion conins wo linrly indpndn prs As wll, h soluion will b consisd of som yp of ponnil funcions Thrfor, ssum h k r is soluion of h sysm, whr k is vcor of cofficins (of nd Subsiu nd r k r ino h quion A, nd w hv r k r A k r Sinc r is nvr zro, w cn lwys divid boh sids by r nd g r k A k W s h his nw quion is cly h rlion h dfins ignvlus nd ignvcors of h cofficin mri A In ohr words, in ordr for funcion k r o sisfy our sysm of diffrnil quions, h numbr r mus b n ignvlu of A, nd h vcor k mus b n ignvcor of A corrsponding o r Jus lik h soluion of scond ordr homognous linr quion, hr r hr possibiliis, dpnding on h numbr of disinc, nd h yp of, ignvlus h cofficin mri A hs 8, Zchry S Tsng D- - 9

20 Th possibiliis r h A hs I Two disinc rl ignvlus II ompl conjug ignvlus III A rpd ignvlu A rld no, (from linr lgbr, w know h ignvcors h ch corrsponds o diffrn ignvlu r lwys linrly indpndn from ch ohrs onsqunly, if r nd r r wo diffrn ignvlus, hn hir rspciv ignvcors k nf k, nd hrfor h corrsponding soluions, r lwys linrly indpndn 8, Zchry S Tsng D- -

21 s I Disinc rl ignvlus If h cofficin mri A hs wo disinc rl ignvlus r nd r, nd hir rspciv ignvcors r k nd k Thn h sysm A hs gnrl soluion r r k k Empl: W hv lrdy found h h cofficin mri hs ignvlus r nd 6 And hy ch rspcivly hs n ignvcor k, k Thrfor, gnrl soluion of his sysm of diffrnil quions is 6 8, Zchry S Tsng D- -

22 8, Zchry S Tsng D- - Empl:, ( Th chrcrisic quion is r r (r (r Th ignvlus r r nd Thy hv, rspcivly, ignvcors For r, h sysm is (A r I (A I Solving h boom quion of h sysm:, w g h rlion Hnc, k, For r, h sysm is (A r I (A I Solving h firs quion of h sysm:, w g h rlion Hnc, k

23 8, Zchry S Tsng D- - Thrfor, gnrl soluion is Apply h iniil vlus, ( Th is W find nd, hnc w hv h priculr soluion

24 s II ompl conjug ignvlus If h cofficin mri A hs wo disinc compl conjug ignvlus λ ± µi Also suppos k b i is n ignvcor (ncssrily hs complvlud nris of h ignvlu λ µi Thn h sysm A hs rl-vlud gnrl soluion λ ( µ bsin( µ ( sin( µ b cos( µ λ cos( A lil dil: Similr o wh w hv don bfor, firs hr ws h compl-vlud gnrl soluion in h form ( λ µ i ( λµ i k k W filr ou h imginry prs by crfully choosing wo ss of cofficins o obin wo corrsponding rl-vlud soluions h r lso linrly indpndn: u v λ λ ( cos( µ b sin( µ ( sin( µ b cos( µ Th rl-vlud gnrl soluion bov is jus u v In priculr, i migh b usful o know how u nd v could b drivd by pnding h following compl-vlud prssion (h fron hlf of h compl-vlud gnrl soluion: k ( λ µ i λ λ ( bi λ ( µ i ( cos( µ isin( µ ibcos( µ i ( cos( µ bsin( µ i λ ( bi(cos( µ isin( µ λ bsin( µ ( sin( µ bcos( µ Thn, u is jus h rl pr of his compl-vlud funcion, nd v is is imginry pr 8, Zchry S Tsng D- -

25 8, Zchry S Tsng D- - Empl: Th chrcrisic quion is r, giving ignvlus r ± i Th is, λ nd µ Tk h firs (h on wih posiiv imginry pr ignvlu r i, nd find on of is ignvcors: (A r I i i Solving h firs quion of h sysm: ( i, w g h rlion ( i Hnc, bi i i k b Thrfor, gnrl soluion is cos( sin( sin( sin( cos( cos( cos( sin( sin( cos(

26 8, Zchry S Tsng D- - 6 Empl: 6, ( Th chrcrisic quion is r r, giving ignvlus r ± i Thus, λ nd µ Tk r i nd find on of is ignvcors: (A r I 6 ( 6 ( i i i i Solving h scond quion of h sysm: ( i, w g h rlion ( i Hnc, bi i i k Th gnrl soluion is sin( sin( cos( cos( sin( cos( cos( sin( sin( cos(

27 8, Zchry S Tsng D- - 7 Apply h iniil vlus o find nd : cos( sin( sin( cos( ( Thrfor, nd onsqunly, h priculr soluion is sin( cos( sin( sin( sin( cos( cos( sin( cos(

28 s III Rpd rl ignvlu Suppos h cofficin mri A hs rpd rl ignvlus r, hr r sub-css (i If r hs wo linrly indpndn ignvcors k nd k Thn h sysm A hs gnrl soluion k r k r No: For mrics, his possibiliy only occurs whn h cofficin mri A is sclr mulipl of h idniy mri Th is, A hs h form α α α, for ny consn α Empl: Th ignvlu is r (rpd Thr r ss of linrly indpndn ignvcors, which could b rprsnd by ny nonzro vcors h r no consn mulipls of ch ohr For mpl k, Thrfor, gnrl soluion is k 8, Zchry S Tsng D- - 8

29 (ii If r, s i usully dos, only hs on linrly indpndn ignvcor k Thn h sysm A hs gnrl soluion k r (k r η r Whr h scond vcor η is ny soluion of h nonhomognous linr sysm of lgbric quions (A r I η k Empl: 7, ( Th ignvlu is r (rpd Th corrsponding sysm is (A r I 7 Boh quions of h sysm r, w g h sm rlion Hnc, hr is only on linrly indpndn ignvcor: k 8, Zchry S Tsng D- - 9

30 8, Zchry S Tsng D- - N, solv for η: η I hs soluion in h form η η η hoos η, w g η / A gnrl soluion is, hrfor, Apply h iniil vlus o find nd Th priculr soluion is

31 Summry: Solving Homognous Sysm of Two Linr Firs Ordr Equions in Two Unknowns Givn: A Firs find h wo ignvlus, r, nd hir rspciv corrsponding ignvcors, k, of h cofficin mri A Dpnding on h ignvlus nd ignvcors, h gnrl soluion is: I Two disinc rl ignvlus r nd r : r r k k II Two compl conjug ignvlus λ ± µi, whr λ µi hs s n ignvcor k b i: λ ( µ bsin( µ ( sin( µ b cos( µ λ cos( III A rpd rl ignvlu r: (i Whn wo linrly indpndn ignvcors is k r k r (ii Whn only on linrly indpndn ignvcor is k r (k r η r No: Solv h sysm (A r I η k o find h vcor η 8, Zchry S Tsng D- -

32 Erciss D-: Rwri h following scond ordr linr quion ino sysm of wo quions y y 6y Thn: ( show h boh h givn quion nd h nw sysm hv h sm chrcrisic quion (b Find h sysm s gnrl soluion 7 Find h gnrl soluion of ch sysm blow Solv h following iniil vlu problms 8, ( 9, (, ( 6 8 8, ( 6 6, ( 8, Zchry S Tsng D- -

33 9, ( 6, (, ( 6 For ch of h iniil vlu problms #8 hrough #, how dos h soluion bhv s? 7 Find h gnrl soluion of h sysm blow, nd drmin h possibl vlus of α nd β such h h iniil vlu problm hs soluion h nds o h zro vcor s α, ( 7 β Answrs D-: ( r 6 r 6, (b 6 7 cos( sin( cos( sin( cos( sin( 8, Zchry S Tsng D- -

34 8, Zchry S Tsng D sin( sin( cos( cos( sin( cos( sin cos sin cos sin( 79 cos( 79 sin( 79 cos( 86 6 For #8 nd 9, ( lim For #,,,, nd, h limis do no is, s ( movs infinily fr wy from h origin For #, 9 ( lim 7 7 ; h priculr soluion will nd o zro s providd h, which cn b chivd whnvr h iniil condiion is such h α β (i, α β, including h cs α β

35 Th Lplc Trnsform Mhod of Solving Sysms of Linr Equions (Opionl opic Th mhod of Lplc rnsforms, in ddiion o solving individul linr diffrnil quions, cn lso b usd o solv sysms of simulnous linr quions Th sm bsic sps of rnsforming, simplifying, nd king h invrs rnsform of h soluion sill pply Empl: 7, ( Bfor w sr, l us rwri h problm ino h plici form of individul linr quions: ( 7 ( W hn firs rnsform boh quions using h usul ruls of Lplc rnsform: sl{ } ( sl{ } L{ } L{ } ( sl{ } ( sl{ } L{ } 7L{ } ( Prilly simplifying boh quions (s L{ } L{ } (* L{ } (s 7L{ } (* 8, Zchry S Tsng D- -

36 Thn muliply q (* by nd q (* by s (s L{ } 6L{ } 8 (** (s L{ } (s (s 7L{ } s (** Subrc q (** from q (** [(s (s 7 ( 6]L{ } s 9 (s 6s 9L{ } s 9 Thrfor, s 9 L{ } ( s s ( s Similrly, muliply q (* by s 7 nd q (* by (s (s 7L{ } (s 7L{ } (s 7 ( 6L{ } (s 7L{ } ( Subrc q ( from q ( [(s (s 7 6]L{ } s 8 (s 6s 9L{ } s 8 s8 L{ } ( s s ( s 8, Zchry S Tsng D- - 6

37 Thrfor, This grs wih h soluion w hv found rlir using h ignvcor mhod Th mhod bov cn lso, wihou ny modificion, b usd o solv nonhomognous sysms of linr diffrnil quions I givs us wy o solv nonhomognous linr sysms wihou hving o lrn spr chniqu In ddiion, i lso llows us o ckl linr sysms wih disconinuous forcing funcions, if ncssry Empl:, ( Rwri h problm plicily nd rnsform: ( ( sl{ } L{ } L{ } s ( sl{ } L{ } L{ } s (6 s Simplify: 8, Zchry S Tsng D- - 7

38 s (s L{ } L{ } s s (* s s L{ } (s L{ } s s s (6* Muliplying q (* by s nd q (6* by, hn subrc h lr from h formr W limin L{ }, o find L{ } s 7s (s s L{ } s s Thrfor, s L{ } s 7s s ( s ( s ( s s s Likwis, muliplying q (* by nd q (6* by s, hn dd hm oghr W find L{ } s s s (s s L{ } s Thrfor, s L{ } s s s ( s ( s ( s s 9 s 9 8, Zchry S Tsng D- - 8

39 Finlly, Ercis D-: Us Lplc rnsforms o solv ch nonhomognous linr sysm, ( 6, ( 8 sin, ( cos 8, ( 6 8, Zchry S Tsng D- - 9

40 8, Zchry S Tsng D- - Answrs D-: 7 7 sin cos sin cos sin cos sin cos

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The Laplace Transform Th Lplc Trnform Dfiniion nd propri of Lplc Trnform, picwi coninuou funcion, h Lplc Trnform mhod of olving iniil vlu problm Th mhod of Lplc rnform i ym h rli on lgbr rhr hn clculu-bd mhod o olv linr diffrnil

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