Linear Systems of Differential Equations

Transcription

1 Linear Sysems of Differenial Equaions A firs order linear n-dimensional sysem of differenial equaions akes he form Y A Y + B, or, in expanded form, y y y n a a a n a a a n a n a n a nn y y y n + b b b n As usual, we define a soluion of his sysem o be a differeniable n-vecor funcion Y which reduces he above o an ideniy upon subsiuion The sysem is homogeneous if B he zero vecor, inhomogeneous oherwise In our discussion we will assume ha he funcions a kj forming he enries of he marix A and he funcions b k forming he componens of he vecor funcion B are a leas piecewise coninuous funcions of he independen variable ; mos examples involve coninuous funcions of Example The sysem of equaions y y y + y y consiues a dimensional linear firs order homogeneous sysem of differenial equaions, < < If we change he sysem o y y y + y + y +

2 we have a dimensional linear firs order inhomogeneous sysem of differenial equaions Here we have A, B The general soluion of a linear homogeneous sysem Y A Y akes he form Y, c, c,, c n c Y + c Y + + c n Y n, where in his formula he Y k, k,,, n, are n-vecor soluions of he sysem; hus y k y k Y k y nk Furher, hese soluions should consiue a fundamenal se of n- vecor soluions, by which we mean ha, given any value of in an inerval a, b in which he sysem saisfies our basic assumpions coninuiy, ec, and given an iniial vecor Y y y y n here is a unique vecor of consans C c c c n such ha, wih Y, c, c,, c n in he form given, Y, c, c,, c n Y If we define a marix Y by specifying is columns o be he soluions Y k ; Y [Y Y Y n ], his is he same hing as saying ha Y C Y,

3 has a unique soluion C for any choice of he vecor Y This is rue, of course, jus in case de Y In ha case we have C Y Y Example In he homogeneous insance of Example given above we can verify ha Y, Y 3 are vecor soluions The general soluion hen akes he form Y, c, c c The corresponding marix Y: Y, + c 3 3 has deerminan de Y 3 which does no vanish in he inerval < <, so we see ha hese form a pair of fundamenal soluions on ha inerval Definiion An n m marix funcion Y whose columns are vecor soluions of he sysem Y A Y is called a marix soluion of ha sysem If Y is n n and he columns are a fundamenal se of soluions, ie, if de Y, hen Y is called a fundamenal marix soluion In eiher case, if we agree ha he derivaive of a marix funcion Y is he marix funcion Y whose enries are he derivaives of he corresponding enries of Y, we have Y A Y 3

4 Proposiion Le Y be an n n marix soluion of he sysem Y A Y on an inerval a < < b where A is coninuous ie, is enries a ij are coninuous here Then he Wronskian deerminan W, Y de Y is eiher idenically zero on a < < b or is never zero on ha inerval Remark Thus he propery of being a fundamenal marix soluion of Y A Y is independen of he choice of in any inerval a < < b where he sysem marix A is a coninuous funcion of The proof will require some properies of deerminans Sup- Proof pose M [M M j M n ] is an n n marix wih columns as indicaed Le ˆM be obained from M by replacing he column M j by ˆM j ; Then ˆM [M ˆMj M n ] i de [M α M j + β ˆM j M n ] α de M + β de ˆM Furher, if ˆMj M k for some k j, hen ii de [M α M j + β M k M n ] α de M If M M is differeniable ie, all of is enries are differeniable, hen iii d de M d n j de [M ˆM j M n] Furhermore, all hree of hese properies remain rue if, insead of working wih he columns of M, we work wih he rows of M 4

5 We will complee he heorem working wih he hree dimensional case; he general n dimensional case is reaed in essenially he same way Thus we suppose ha we have he sysem y y y 3 and hree soluion vecors Y j a a a 3 a a a 3 a 3 a 3 a 33 y j y j, j,, 3 y 3j y y y 3 forming he columns of a 3 3 marix soluion Y Differeniaing de Y by rows we have suppressing now for breviy d de Y d de y y y 3 y y y 3 y 3 y 3 y 33 y y y 3 y y y 3 + de y y y 3 + de y y y 3 y 3 y 3 y 33 y 3 y 3 y 33 Looking a jus he firs of hese hree marices and using he differenial equaions implied by Y A Y we have y y y 3 y y y 3 y 3 y 3 y 33 a y + a y + a 3 y 3 a y + a y + a 3 y 3 a y 3 + a y 3 + a 3 y 33 y y y 3 y 3 y 3 y 33 In he firs row he second and hird erms of each enry are jus a imes he corresponding enries of he second row of Y plus a 3 5

6 imes he corresponding enries of he hird row of Y Using he row versions of i and ii, we ge jus a de Y The oher wo marices in he formula for de Y, manipulaed in he same way, yield a de Y and a 33 de Y Thus he final resul becomes d de Y d a + a + a 33 de Y T r A de Y The race of a square marix M is he sum of is diagonal enries and is wrien T r M This is a firs order scalar linear homogeneous equaion and we hus have de Y exp T r As ds de Y for any values of and in he inerval a, b Since he exponenial funcion is never zero, we see ha de Y if and only if de Y and he proposiion follows from his Proposiion If Y is an n n marix soluion for Y A Y and C is a consan n m marix, hen Y C is an n m marix soluion for Y A Y Remark: C could be an n marix; ie, a column vecor C; hen Y C is a vecor soluion Proof We jus muliply he marix equaion Y A Y on he righ by C and use Y C YC Now suppose we wan o find he soluion of an iniial value problem Y A Y, Y Y, where is a given value of he independen variable and Y is a given vecor Suppose we have a fundamenal marix soluion Y; ie, one 6

7 for which de Y Le us ry o find a soluion of he iniial value problem in he form Y Y C, where C is a consan column vecor Our proposiion shows ha Y, hus defined, is a soluion Evaluaing a we have Y Y C Y C Y Y From his we see ha he iniial value problem has he unique soluion Y Y Y Y Thus we can solve iniial value problems wih ease once we have a fundamenal marix soluion Example 3 Le us again consider he sysem y y y + y y Suppose we wan he soluion corresponding o y, y From our earlier example Y 3 is a marix soluion; since is deerminan is 3, i is a fundamenal marix soluion for > Accordingly, he soluion of he given iniial value problem a is given by Since y y 3 3 he soluion is y 3 y

8 The fac ha he soluion of he iniial value problem Y Y can be wrien in he form Y Y Y Y for a e, any fundamenal marix soluion Y jusifies referring o Y C, for an arbirary n-vecor C, as a general vecor soluion for he sysem Y A Y ; he choice C Y Y yields Y wih Y Y We have seen ha for a marix soluion Y and an n m consan marix C, Y C is also a marix soluion If Y is a fundamenal marix soluion, hus n n and nonsingular for each under consideraion, hen, given, Y exiss and Y, YY is again a fundamenal marix soluion This paricular fundamenal soluion has he special propery Y, I, he n n ideniy marix In general, for arbirary, τ, Y, τ is he fundamenal marix soluion such ha Yτ, τ Y, I Example 4 For he sysem in Example 3 we have Y 3 for which he inverse marix a τ and Y, τ are, respecively, Yτ τ τ τ, Y, τ YYτ τ 3 τ τ 8