After the completion of this section the student. Theory of Linear Systems of ODEs. Autonomous Systems. Review Questions and Exercises
|
|
- Maria Baldwin
- 6 years ago
- Views:
Transcription
1 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 V.5 SYSTEMS OF FIRST ORDER LINEAR ODEs Objecives: Afer he compleion of his secion he suden - should recall he definiion of a sysem of linear s order ODEs - should be able o find he general soluion of a sysem wih consan coefficiens - should be able o invesigae for sabiliy he equilibrium poin of a plane sysem Conens: V.5.. Definiions and Noaions V.5.. V.5.3. V.5.4. V.5.5. V.5.6. V.5.7. Theory of Linear Sysems of ODEs The Fundamenal Se of a Linear Sysem wih Consan Coefficiens Auonomous Sysems Examples Review Quesions and Exercises Sysems of ODEs wih Maple
2 46 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions V.5 SYSTEMS OF s ORDER LINEAR ODEs. DEFINITIONS AND NOTATIONS In his secion we will sudy he heory of he sysems of linear s order ODEs. I can be shown ha such sysems are equivalen o a single linear differenial equaion of a higher order; and for boh of hem he mos heoreical resuls have a similar descripion. Alhough, we will ry o avoid duplicaion of he heoreical jusificaion, in a pracical approach, he mehods of soluion for a single equaion and for he sysems are differen. In many cases, he descripion of he physical model is more naural o perform wih he sysems of ODEs, and invesigaion of he physical models such as dynamic, sabiliy ec. is more visual when i is made wih he help of sysems. Normal form Consider a sysem of n linear firs order ODEs wrien in he normal form (solved for he derivaives of unknown funcions): ( ) x + a ( ) x + + an ( ) xn f( ) ( ) x + a ( ) x + + a ( ) x f ( ) x = a x = a + n n + () x = a x + a x + + a x f where x ( ), x ( ),, xn ( ) sysem (), and coefficiens ( ) n n n nn n + are unknown funcions o be deermined from he a ij,, j =,,... f i are coninuous funcions in D R. i and funcions n Marix form Inroduce he following column vecors and a marix wih enries which are he elemens of he sysem (): x = x x x n ( ) ( ) ( ) x x x = x n ( ) ( ) ( ) f f f = f n ( ) ( ) ( ) a a an a a an A = an ( ) an ( ) ann ( ) Then he sysem () can be wrien in he compac marix form: x = Ax + f () This is a non-homogeneous sysem. Wihou a free erm f, he sysem is homogeneous: x = Ax (3) Soluion vecor The soluion vecor (paricular soluion) is any column vecor x ( ), D which saisfies he sysem (). Iniial value problem Find he soluion of a sysem of ODEs subjec o iniial condiions: x x x = Ax + f subjec o x( ) = x where x =, x i (4) x n
3 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 47 Exisence Theorem Theorem (exisence and uniqueness heorem) Le he sysem of linear s order ODEs x = Ax + f be normal for D (means ha he funcions ( ) and f i ( ) are coninuous on D ) and le xi. Then here exis exacly one soluion x ( ) such ha x ( ) x x = x n D a ij The paricular soluion x ( ) of he sysem () or (3) is a vecor valued funcion n ( ) : x n which can be reaed as a parameric definiion of he curve in he space : x = x( ) x = x( ) x( ) = D (5) xn = xn( ) wih he coordinaes x,x,...,x n. phase porrai rajecory phase space This space is called a phase space, and he soluion curve defined paramerically by equaion (5) is called he rajecory or he orbi in he phase space. For -D and 3-D cases, he radiional noaions for he coordinae sysem and, correspondingly, for unknown funcions in he sysem are used: x = a x+ a y+ a z+ f 3 y = a x+ a y+ a z+ f (b) 3 z = a x+ a y+ a z+ f wih he paricular soluions wrien as x= x x ( ) = y = y D (5b) z = z The graph of he equaion (5b) defines a rajecory in he phase space (phase plane, for -D case). The independen variable D can be reaed as he ime (can also be negaive), and for any momen of ime equaion (5b) defines he posiion of a poin on he rajecory herefore, he parameric equaion (5b) can be inerpreed as a moion of a maerial poin along he rajecories defined by he linear sysem of ODEs. The arrows on he rajecories indicae he direcion of moion wih he increase of ime. The family of all rajecories of he linear sysem is called he phase porrai. An iniial value problem defines he rajecory which goes hrough he prescribed poin. According o he Exisence Theorem, he soluion of any iniial value problem of he linear sysem is unique i means ha here is only one rajecory which goes hrough any poin of he phase space, and ha he rajecories of he linear sysem do no inersec.
4 48 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions. THEORY OF LINEAR SYSTEMS OF ODEs I can be shown ha a linear n h order ODE can be ransformed o a sysem of n linear s order ODEs, and a sysem of n linear s order ODEs can be reduced o a linear n h order ODE. Therefore, he descripion and he properies of he general soluion of he sysem will be similar o hose of he general soluion of a linear n h order ODE. Here, we will lis he major resuls of he heory of he sysems of linear s order ODEs. I) Homogeneous Sysem: x = Ax (3) linear independen soluions Vecors (vecor funcions) ( ), ( ),..., ( ) x x x are linearly independen if n heir linear combinaion equals zero for all D c x + c x c x = (6) n n only if all coefficiens are zero c = c =... = cn =. If vecors ( ), ( ),..., ( ) x x x are no linearly independen, hen hey are n linearly dependen. I means ha in he linear combinaion a leas one coefficien ck can be non-zero. Wronskian The Wronskian of he se of soluion vecors of he homogeneous sysem x, x,..., x is defined as a deerminan of he marix whose columns n are he vecors x k ( ) : W ( ) = de ( ) ( )... ( ) x x x n (7) There is a relaionship of he Wronskian (7) o he Wronskian defined in 5.3. If W( ) a leas a one poin D, hen he soluion vecors x ( ), x ( ),..., x ( ) are linearly independen. n There always exis n linear independen soluions ( ), ( ),..., ( ) x x x of he n homogeneous sysem (3). Bu any n+ soluions of he homogeneous sysem (3) are linearly dependen. Fundamenal se Any se of n linearly independen soluions of he sysem (3) x, x,..., x is called a fundamenal se. n I is obvious ha homogeneous sysems always possess a zero soluion x( ) (rivial soluion). Bu any se which includes he zero vecor is linearly dependen. Therefore, he fundamenal se canno include he rivial soluion. Fundamenal marix A marix wih columns which are consruced from he fundamenal se is said o be he fundamenal marix: ( ) = ( ) ( )... ( ) X x x x n (8) General Soluion Any soluion of he homogeneous sysem (3) can be wrien as a linear combinaion of he vecors from he fundamenal se. Therefore, he general soluion (complee soluion, complimenary soluion) of he homogeneous sysem is a se of all is soluions and i is given by all linear combinaions of he
5 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 49 vecors from he fundamenal se (span of he fundamenal se) and i can be defined as: ( ) c ( ) c ( )... c ( ) x = x + x + + x = Xc (9) c n n where c is a vecor of arbirary consans. Equaion (9) defines a family of curves in he phase space which represens he soluions of he homogeneous sysem. Soluion of IVP The soluion of he iniial value problem for a homogeneous sysem: = x = x () is given by x Ax subjec o = x X X x () where X ( ) is he fundamenal marix and X is he inverse of he fundamenal marix a = II) Non-Homogeneous Sysem: x = Ax + f () Denoe by ( ) x any paricular soluion of he sysem (). A paricular soluion p can be found by he mehod of undeermined coefficiens (similar o he case of linear ODE) or by he mehod of variaion of parameer: xp = X X f d () The general soluion of he non-homogeneous sysem () is given by a sum of he general soluion of he homogeneous sysem (complemenary soluion) and a paricular soluion: x ( ) = x ( ) + x ( ) c p General Soluion Using equaions (9) and (), he general soluion of he non-homogeneous sysem can be wrien as: x ( ) = X( ) c + X X f d (3a) Soluion of IVP The soluion of he IVP for he non-homogeneous equaion (4) can be given by: x ( ) = X X x + X X s f s ds (3b) In a conclusion, he general soluion and he soluion of IVP for homogeneous and non-homogeneous sysems (9), (), () and (3) can be deermined if he X is known. In he nex fundamenal marix of he homogeneous sysem secion we will consider he case of linear sysems wih consan coefficiens (auonomous sysems) for which here exis he developed mehods of finding he fundamenal marix.
6 4 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 3. FUNDAMENTAL SET OF LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS Consider he homogeneous sysem of linear s order ODEs x = ax + ax + + an xn x = a x + a x + + a x n n (4) x = a x + a x + + a x n n n nn n where all coefficiens aij, i, j =,,... are consans. In marix form: a a an a a a n x = Ax A = (5) an an ann The Eigenvalue Problem, marix diagonalizaion, marix exponenial are among he echniques used for consrucion of he fundamenal se for a sysem wih consan coefficiens. Eigenvalue Problem: Because he linear sysem is of he firs order, we look for he non-rivial soluion of he exponenial form k x( ) = k e λ, k = (6) k n k is a non-zero (non rivial) vecor of consans, where k i and λ can be he real or he complex numbers which have o be found from saisfying equaion (5). Subsiue he rial form (6) ino marix equaion (5): ( k λ e ) = A λ ( k e ) λ λ e = k Ak e λ λ λ λ e e = k Ak Ike Ake I is he ideniy marix λ λ λ = e λ ( λi A) k = ( λ ) = λi A k = can be divided by e λ > A I k (7) This is he homogeneous sysem of algebraic equaions, which according o he Theorem has a non-rivial soluion if he deerminan of he marix of coefficiens is equal o zero. Therefore, he following condiion should be saisfied: de A λi = (8) Equaion (8) is he de h n order algebraic equaion for consan λ : ( A λi ) a a an a a a n de λ = an an ann a λ a an a a λ a n de = an an ann λ = c λ + c λ c λ+ c n n n n
7 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 4 Expansion of he deerminan yields an algebraic equaion wih real coefficiens which is called he characerisic equaion. According o he Fundamenal Theorem of Algebra i has n roos λ, λ,..., λn which can be real or complex, disinc or repeaed. These roos are called he eigenvalues. Afer he eigenvalues are deermined, hey can be subsiued ino equaion (7) and he corresponding non-zero soluions k, k,..., k n of he vecor equaion can be found. These soluions k, k,..., k n are called he eigenvecors corresponding o eigenvalues λ, λ,..., λ n. The soluion of he eigenvalue problem (7) is no unique; hey can be chosen in such a way ha he desired soluions (6) have only he real-valued componens. Then he consruced fundamenal marix also will have only real-valued enries. Le us show how i can be done: Fundamenal Marix: Case : All eigenvalues λ, λ,..., λ n are real and disinc. Then he corresponding eigenvecors vecors k, k,..., k n are also real-valued and linearly independen. Therefore, he fundamenal marix can be defined as n ( ) e λ e λ... e λ X k k k (9) = n Exercise: show ha he Wronskian is no equal o zero for any. In general, for real disinc eigenvalues λ, λ,..., λ m he corresponding enries of he fundamenal marix are λ λ m k k k () e e... e λ m Case : Le eigenvalue λ be he repeaed real roo of he characerisic equaion (8) of mulipliciy m. Then if: a) here are m linearly independen eigenvecors k, k,..., k m corresponding o he eigenvalue λ. Then he fundamenal marix includes m ke λ ke λ... k me λ b) here is only one linearly independen eigenvecor k corresponding o he eigenvalue λ. Then he oher linearly independen soluions can be consruced in he following way: find he vecors kpq,,,... which are a soluion of he vecor equaions A λi k = ( λ ) ( λ ) A I p = k A I q = p Then he fundamenal marix will include he vecors ke λ, ke λ + pe λ, k e λ + pe λ + q e λ,... () I can be verified wih he help of he Wronskian ha hese vecors are linearly indeneden. Case 3: The eigenvalue λ is complex. We know ha he complex roos of he algebraic equaion wih real coefficiens appear in conjugae pairs: λ = α ± β, i The corresponding eigenvecors are also complex conjugaes k and k. Therefore for disinc eigenvalues λ, = α ± βi here are wo linearly independen soluions e ( α+ βi ) k and k e ( α βi ). Bu
8 4 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions hey are complex-valued funcions which is no convenien for represenaion of he physical problems. I can be shown ha he linear combinaions of hese wo soluions and applicaion of he a+ bi a e = cos b + i sinb e ) yield he wo Euler formula ( independen real-valued soluions which can be included in he fundamenal marix: ( cos β sin β) e α x = b b ( cos β sin β) e α x = b + b () where vecors are b = Re( k ) and = Im b k. Conclusion: The soluion of he eigenvalue problem for he homogeneous linear sysem of s order ODEs wih consan coefficiens yields he fundamenal marix X. Marix exponenial: The sysem of s order ODEs in marix form x = Ax resembles a s order ODE for which i is very emping o wrie he soluion in he radiional exponenial form e A. Bu how can an exponenial funcion wih he marix be calculaed? Again, we can use he analogy wih he calculus of funcions of a single variable and define he marix-valued exponenial funcion in he form of he Taylor series: k 3 A A 3 e = = k= k!! 3! I A A A (3) k imes in which we know how o calculae he powers of he marix A k = AA A (i can be shown ha he infinie series (3) is always convergen for any ). Then he fundamenal marix of he linear sysem (5) can be wrien as X (3) = e A Then he general soluion in marix exponenial form is A A A x = e c+ e e f d (4) The soluion of he IVP can be defined by ( ) A A sa x = e x + e e f s ds (5) The marix exponenial form of he fundamenal marix is no used very ofen for he acual soluion of a linear sysem of ODEs. Bu i is very convenien for derivaion and proofs of he heoreical resuls such as exisence heorems ec.
9 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions AUTONOMOUS SYSTEMS: A sysem of s order ODEs is called auonomous if i can be wrien in he form: x = f x,x,...x n x = f x,x,...x n x = f x,x,...x n n n (6a) and in a marix form: x = f x (6b The unknown funcions x( ),x( ),...xn( ) are funcions of, bu he independen variable does no appear explicily in he righ hand side of he sysem (6). Auonomous sysems are no necessarily linear. plane linear auonomous sysems Here, we will consider only plane linear auonomous sysems, which for simpliciy can be wrien as x = ax + by y = cx + dy a,b,c,d < < (7a) and in marix form x = Ax (7b) The paricular soluion of he plane sysem is a -dimensional vecor which paramerically describes a rajecory (orbi) on he phase plane: x ( ) x = y x= x y = y < < (8) The general soluion also includes an arbirary consan vecor ( ) x,c x(, c ) = (9) y,c I defines he family of rajecories in he phase plane (phase porrai) and describes he moion of he poins along he soluion curves wih he change of ime. The arrows on he rajecories indicae he direcion of he moion of he poin wih he increase of ime. This mapping defined by equaion (9) is called a dynamical sysem. x x (, x ) The iniial value problem = x Ax, x(,x ) (, ) y(,y ) x = x has a unique soluion x = x x = he rajecory which goes hrough he prescribed poin x. The righ hand side of he vecor equaion (7b) defines he vecor field in he phase plane. A any poin on he plane x we can draw a vecor Ax and obain a geomerical represenaion of he direcional field. These direcional vecors are angen o he rajecories defined by he sysem (7b). The direcional vecor field can be drawn even wihou solving sysem (7), bu i provides a qualiaive picure of he dynamical sysem.
10 44 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions Criical Poins: Sabiliy of criical poins: sable The imporan characerisics of he auonomous sysems are he criical (equilibrium, saionary) poins, which can be defined as he soluions no changing in ime (consan soluions): if a poin is placed a he equilibrium poin i will remain here forever. The criical poins can be defined as he soluions of he equaion: f x = For he plane sysem, criical poins are he soluions of he sysem of equaions: ax + by = cx + dy = If he deerminan a b de = ad bc c d here is only one criical poin x cr = (,) (isolaed criical poin). If he deerminan a b de = ad bc = c d ad = bc hen here are infiniely many criical poins which are locaed on he line a y = x b These criical poins are no isolaed. For a non-linear dynamical sysem, he siuaion wih he criical poins is more complicaed. Le cr f x x be he isolaed criical poin of he auonomous sysem (7):. x cr is said o be sable if for any neighborhood ( cr ) smaller neighborhood V ( x cr ) such ha for any V ( cr ) x(,x ) = (, ) = U ( cr ) y(,y ) cr = U x here exiss a x x he rajecory x x x x for all I means ha he rajecory which sars in V remains compleely in U. unsable. x cr is said o be unsable if i is no sable. I means ha i does no maer how close o he criical poin he saring poin x will be, some rajecory will go ouside of any neighborhood U ( xcr ) of he criical poin. asympoically sable 3. x cr is said o be asympoically sable if a) x cr is sable; b) here exiss a neighborhood W ( x cr ) such ha lim x(, x ) = x cr Here, we will invesigae he sabiliy of he plane dynamical sysems (7) which can have only one isolaed criical poin x cr = (,) depending on he marix of coefficiens A.
11 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 Phase Porrai of he Plane Sysem: x = ax + by y = cx + dy a b A = c d a b de A = de = ad bc c d Eigenvalue Problem: The characerisic equaion: a λ b A λi = = λ ( + ) λ+ = λ λ+ = c d λ A A de a d ad bc Tr de ( a + d ) ± ( a + d ) 4 ( ad bc) Eigenvalues: λ, = The form of eigenvalues depends on he expression under he square roo which is called he discriminan: Discriminan: ( A) = a + d 4 ad bc = Tr 4 de A I) If > hen he eigenvalues are real and disinc λ λ II) If = hen he eigenvalues are real and repeaed λ = λ = λ III) If < hen he eigenvalues are complex conjugaes λ, = α ± βi Consider he possible configuraions of he plane phase porrai (for simpliciy of presenaion, he deails of soluion will be skipped; derivaion of some of he resuls will be conduced in he examples and he exercises) : λ λ > General soluion: I) x = c k e + c k e a) λ λ >, > Boh eigenvalues are posiive Unsable node b) λ λ <, < Boh eigenvalues are negaive λ λ λ λ lim x = lim c k e + c k e = c k lim e + c k lim e = Sable node c) λ λ >, < The eigenvalues are of he opposie sign Saddle poin (unsable)
12 46 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions II) = λ = λ = λ λ > a) There are wo linearly independen eigenvecors k, k c e c e c c e General soluion: x = k + k = ( k + k ) λ λ λ i) λ > degenerae (proper) unsable node ii) λ < degenerae (proper) sable node λ > b) There is one linearly independen eigenvecor k (find p, q,... ). General soluion: λ λ c c λ x( ) = cke + c( k + p) e = k+ p+ ck e i) λ > degenerae (improper) unsable node ii) λ < degenerae (improper) sable node III) < λ, = α ± βi eigenvecors k = b + i b, k = b ib α > a) α General Soluion: x = c ( bcos β b sin β) + c( bcos β + b sin β) e α i) α > unsable focus (spiral poin) α < ii) α < sable focus (spiral poin) asympoically sable b) α =, λ, = ± βi (pure imaginary, when a = d ) α = General Soluion: ( b β b β ) ( b β b β ) x = c cos sin + c cos + sin sable cener (no asympoically sable)
13 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 47 Classificaion of he criical poins of he plane linear sysem sable degenerae node sable focus de A sable cener unsable focus = de A = ( TrA) 4 unsable degenerae node sable node unsable node TrA saddle poin Procedure for Soluion of he Linear Sysem of s Order ODEs wih Consan Coefficiens:. Wrie he sysem in he normal marix form (): x = Ax + f. Solve he Eigenvalue Problem (7): A λi k = o find eigenvalues λ i and eigenvecors 3. Consruc he fundamenal marix (8) X in correspondence wih equaions (9-). k i. 4. Calculae he general soluion according o he variaion of parameer formula (3a): x ( ) = X( ) c X X f + d 5. For soluion of he IVP (4) wih x = x, use he variaion of parameer formula (3b): x ( ) = X X x X X f + s s ds
14 48 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 5. EXAMPLES: ) (reducion of he sysem of s order ODEs o a higher order ODE) Consider he sysem of wo s order ODEs: x = 3x y y = x y Reducion is performed by differeniaion of he equaions and consecuive replacemen of he unknown funcions unil a differenial equaion for a single unknown funcion is obained. Consider he second equaion: y = x y x = y + y ( ) Differeniae he second equaion wih respec o y = x y x = y + y Subsiue expressions for x and x ino he firs equaion x = 3x y y + y = 3 y + y y Rearrange i ino he equaion for he funcion y y y + y = This is a single nd order ODE, linear homogeneous wih consan coefficiens, which can be solved by he sandard mehod: auxiliary equaion general soluion m m + = m, = y c e c e The funcion x can be found from equaion ( ): x y y = ce + ce + ce + ce = c + c + e = + = ( ce + ce ) + ( ce + ce ) Therefore, he general soluion of he sysem of equaions is: = + y c e c e x c c e = + + = + ( ) Tha can be verified by direc subsiuion ino he original sysem of equaions.
15 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 49 ) (reducion of a higher order ODE o a sysem of s order ODEs) h Consider a normal n order linear ODE = ( n) ( n ) n a x y a x y... a x y f x Divide he equaion by a ( x) : ( n) ( n ) n y y... y a x for all x ( ) a x a x f x = + ( ) a x a x a x Inroduce he se of new funcions: x = y x = y x = x x3 = y x = x3 xn Differeniae x n ( n ) y = x n = xn ( n) x n = y Wih hese noaions, he equaion ( ) can be rewrien as a( x) an( x) f ( x) x n = x n... x + a x a x a x Collec hese in he normal linear sysem of s order ODEs: x = x x = x3 x = x n n n n n a x a x f x x = x... x + ( ) a x a x a x 3) Solve he nd order ODE y y + y = by reducion o a sysem of s order ODEs. Applying ( ) for he nd order equaion, we obain x = x a x a x f x x = x x + = x x a x a x a x (Surprisingly, his sysem is no idenical o he sysem of Example ) In marix form: x x x = x Find he fundamenal se for a sysem wih consan coefficiens. Characerisic equaion (8): de λ = λ λ + = λ There is only one eigenvalue λ = of mulipliciy. Find eigenvecors by plugging λ = in vecor equaion (7): k = k
16 4 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions k = k I is a singular linear sysem of algebraic equaions, i has only one independen soluion: k = Find he vecor p by soluion of he equaion ( λ ) A I p = k p = p Then p,p can be found from he equaion p + p = One of he soluions can be p = Then he fundamenal marix is: λ λ λ X= ke ke + p e e [ ] X= k k + p X = e + Then he general soluion of he sysem is given by x = Xc or in he componen form = + = + ( + ) x c e c e x c e c e This is he soluion of he sysem of ODEs o which he ODE was reduced. Recall now ha x ( ) was defined as x y ( ) = x ( ) = c e + c e = y, herefore, he general soluion is Which coincides wih he previously obained general soluion ( ). The second soluion can be reaed as y = x = c e + c + e Inegraion of his equaion will duplicae he previous resul.
17 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 4 4a) (Linear Sysem of equaions General Soluion) Find he general soluion of he sysem of ODEs: Soluion: x = x + x + e 3 x = 4x + 3x + 8e ) Rewrie he given sysem in he marix form: 3 x e = x 8e x x ) Solve he eigenvalue problem: k λ 4 3 = k Characerisic equaion: λ 4 3 = = = 4 3 λ Eigenvalues: λ =, λ = 5 (real disinc) de λ λ 4λ 5 Eigenvecors: λ = k λ 4 3 = k k = k k 4 4 = k k 5 k λ = = 5 k 4 k = 4 k 3) Fundamenal marix: 5 e e X ( ) = 5 e e e e k = = X ( ) = de = e e = 3e 5 e e 4) Variaion of parameer formula (3a): Inverse of he Fundamenal marix: e e 3 3 X ( ) = 5 5 e e 3 3 e e e 8e + 6e X ( ) f ( ) = = e 4e + 6e e e 3 3
18 4 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions ( ) X f Paricular soluion: e d + 6 e d 4 e + 3 e 3 e d = = 3 e 4 e d + 6 e d e e e + e 4e xp = X( ) X ( ) f ( ) d = 5 3 = 3 e e e e 6e e Complimenary Soluion x c : Complimenary Soluion (soluion of he homogeneous sysem): k = k = 5 5 e e c ce + ce xc = X( ) c = 5 5 e e c = ce + ce General soluion: saddle poin (eigenvalues of opposie sign) x ( ) = x ( ) + x ( ) c p 5 ce + ce 4e = 3 3 ce + ce 6e e 4b) (Linear Sysem of equaions Iniial Value Problem) Find he soluion of he sysem of ODEs: x = x + x + e 3 x = 4x + 3x + 8e subjec o he iniial condiion: x ( ) 3 = Soluion: Use he fundamenal se of he previous example. 3) Fundamenal marix: 5 e e X ( ) = 5 e e 4) Variaion of parameer formula (3b): X X ( ) ( ) e e 3 3 = 5 5 e e = 3 3
19 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 43 X X x 5 5 e e 3 33 e + e = 5 5 e e = e + e 3 3 ( ) ( ) X f e e e 8e + 6e = = e 4e 6e e e = = = 4s 3s e ds + 6 e ds e e 4 3 e e ( s) ( s) ds + X f 3 3 s 3s e e e e e ds + 6 e ds e e e + e 4e + 4e X( ) X ( s) f ( s) ds = 5 3 = 3 5 e e e e + 4 6e e + 8e Soluion of IVP: x ( ) e + e 4e + 4e e 4e + 5e = = 3 5 e + e 6e e + 8e e 6e e + e x( )
20 44 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 5) (sabiliy of auonomous sysem) Invesigae for sabiliy he equilibrium poin and skech he phase porrai of he following auonomous sysem: x = x+ 3y y = 3x y Soluion: ) Rewrie he given sysem in marix form: x 3 x y = 3 y ) Solve he eigenvalue problem: 3 k λ = 3 k Characerisic equaion: 3 λ λ de λ = = λ + 4λ+ 3 = Eigenvalues: λ = + 3i, λ = 3i (complex) Eigenvecors: λ = + 3i 3 k λ = 3 k + 3i k 4 3 = + 3i k 3i 3 k = 3 3i k k = i i k = = + i 3) Fundamenal marix (use equaion ()): α x = b cos β b sin β e = cos 3 sin 3 e b =, b = x = b + b = + X ( ) α ( cos β sin β ) e cos 3 sin 3 e cos 3 e sin 3 e = sin 3 e cos 3 e 4) General soluion: = ( ) x X c cos 3 e sin 3 e c sin 3 e cos 3 e c = c cos 3 e + c sin 3 e = csin 3 e + c cos 3 e
21 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 In parameric form: = + = + x c cos 3 e c sin 3 e y c sin 3 e c cos 3 e According o case III) a) ii) his is an asympoically sable focus. Skech he phase porrai: For he paricular curve, choose c =, c =, hen = = x y cos 3 e sin 3 e A graphing calculaor can be used for skeching he graph of his curve, bu i is imporan o know how o skech he graph jus from he parameric equaion we can perform i qualiaively in he following way: For = x = y = The saring poin is defined. Now le us see where he curve will go under a small increase of ime x = ε = ε y = ε Then coninue he curve as a shrinking spiral in he deermined direcion: All oher rajecories will be of he same shape, covering he enire plane wihou inersecions. Here is he graph generaed by Maple: Conclusion: he equilibrium poin is he asympoically sable focus (spiral poin).
22 46 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 6. REVIEW QUESTIONS AND EXERCISES: ) Wha is a sysem of differenial equaions? ) Wha ype of sysems did we sudy in his secion? 3) How many soluions of a normal sysem of s order ODEs can go hrough an arbirary poin of he plane? 4) How many soluions of a homogeneous sysem of s order ODEs can go hrough he poin (, )? 5) Why is uniqueness no violaed for a saddle equilibrium poin? 6) Why is he sysem of wo s order ODEs called a dynamical sysem? 7) Wha is he sabiliy of an equilibrium poin? Wha does i mean? EXERCISES: ) Reduce he following ODEs o a sysem of s order ODEs: a) b) y + 5y + 3y 6 y = e ( iv) y 6y + y y 3y = cos Reduce he sysem of s order ODEs o a higher order ODE: c) x = 4x + x + x = x + 3x ) Marix exponenial: a) Using he definiion of marix exponenial, verify he differeniaion rule: d e d A = A e A b) Show ha if a a A = ann hen e A a e a e = ann e c) Consider IVP: solve = x = x. x Ax subjec o A Show ha e = X( ) X ( ), where ( ) X is he fundamenal marix. 3) Find he general soluion of he following sysems and skech he soluion curves of he homogeneous par of he sysems: a) x x = b) x = 5x + 3x + e x = x + sec x = x x + c) x = 4x + x + d) x = x + x + sin x = x + 3x x = x + cos
23 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 47 e) x = x + x f) x = 3x x x3 x = x + 4x3 x = x + x x3 + x 3 = x 4x3 x = x x + x + e 3 3 4) Find he soluion of he following Iniial Value Problems and skech he graph of he soluion: a) = + b) x 3x x 4e x = 3x x + 4 = + + x x 3x 4e x = 5x 3x + 3 subjec o x ( ) =, x ( ) = ( π ) ( π ) x =, x = c) x = 3x x x3 c) x = 3x x x3 x = x + x x3 + x = x + x x3 + x = x x + x + e 3 3 x = x x + x + e 3 3 = = = x, x, x 3 x =, x =, x = 3 5) Invesigae for sabiliy he equilibrium poin and skech he phase porrai of he following auonomous sysems: a) x = x+ 3y b) x = x+ 4y y = 3x+ y y = x+ y c) x = x+ 5y d) x = x 4y y = x y y = x+ 5y e) x = x+ y f) x = x+ y y = x+ y y = x+ y g) x = x y h) x = x α y y = x+ 4y y = x y
24 48 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 7. LINEAR SYSTEMS OF ODEs WITH MAPLE. a) Find he general soluion of he homogeneous sysem and skech he phase porrai: x = x x x = 3x x + 4 > wih(linalg): > k:=marix(,,[[],[]]); k := > f:=marix(,,[[],[4*]]); f := 4 > C:=marix(,,[[c[]],[c[]]]); C := Eigenvalue Problem: > A:=marix(,,[[,-],[3,-]]); c c A := > eigenvecs(a); [,, {[, ]}], [-,, {[, 3 ]}] Fundamenal marix: > X:=marix(,,[[exp(),exp(-)],[exp(),3*exp(-)]]); X := e e ( ) e 3 e ( ) Complimenary Soluion - General Soluion of Homogeneous Sysem: > Xc:=evalm(X&*C); e c Xc := + e ( ) c e c + 3 e ( ) c -Phase Porrai: > x():=exp()*c[]+exp(-)*c[]; x( ) := e c + e ( ) c > y():=exp()*c[]+3*exp(-)*c[]; y( ) := e c + 3 e ( ) c > p:={seq(seq(subs({c[]=i/*,c[]=j/*}, [x(),y(),=-4..4]),i=-4..4),j=-4..4)}: > plo(p,x=-5..5,y=-..,color=black);
25 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 49 b) Find he general soluion of he non-homogeneous sysem: Paricular Soluion - Variaion of Parameer: > Xinv:=simplify(inverse(X)); Xinv := > simplify(evalm(xinv&*f)); > map(in,%,); 3 ) e( ) e( e e e ( ) e e ( ) + e ( ) e e > Xp:=simplify(evalm(X&*%)); General Soluion: > GS:=evalm(Xc+Xp); Xp := e c GS := + e ( ) c + 4 e c e ( ) c 8 4 Soluion Curves: > xn():=exp()*c[]+exp(-)*c[]+4*; xn( ) := e c + e ( ) c + 4 > yn():=exp()*c[]+3*exp(-)*c[]+8*-4; yn( ) := e c + 3 e ( ) c > pn:={seq(seq(subs({c[]=i/,c[]=j/}, [xn(),yn(),=-..]),i=-3..3),j=-..)}: > plo(pn,x=-8..6,y=-..,color=black, numpoins=5);
26 43 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions c) Find he soluion of he Iniial Value Problem: Soluion of IVP - Variaion of parameer formula (3b): > Xinv:=simplify(subs(=,evalm(Xinv))); Xinv := > X:=evalm(evalm(X&*Xinv)&*k); X := 3 e e ( ) 3 e 3 e ( ) > X:=simplify(evalm(Xinv&*f)); X := e ( ) e > X3:=subs(=s,evalm(X)); X3 := e ( s ) s e s s > X4:=simplify(map(in,X3,s=..)); X4 := e ( ) + e e e + > X5:=simplify(evalm(X&*X4)); > XS:=evalm(X+X5); Graph of he soluion of IVP: X5 := 4 e + e ( ) 8 4 e + 6 e ( ) XS := e + e ( ) + 4 e + 3 e ( ) > u:=exp()+exp(-)+4*; u := e + e ( ) + 4 > v:=exp()+3*exp(-)+8*-4; v := e + 3 e ( ) > plo([u,v,=-3..]);
27 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 43 Applicaion of he sandard Maple procedure for soluion of he sysem of ODEs.. Invesigae for sabiliy he equilibrium poin and skech he phase porrai of he following auonomous sysem: x = x+ 3y y = 3x+ y > eq:=diff(x(),)=-4*x()+3*y(); d eq := x( ) = 4 x( ) + 3 y( ) d > eq:=diff(y(),)=-*x()+*y(); d eq := y( ) = x( ) + y( ) d > Soluion:=dsolve({eq,eq},{x(),y()}); > assign(soluion): Soluion := { x( ) = _C e ( ) + _C e ( ), y( ) = _C ) e( + _C e ( ) } 3 > p:={seq(seq(subs({_c=i/,_c=j/}, [x(),y(),=-..]),i=-5..5),j=-5..5)}: > plo(p,x=-3..3,y=-3..3,color=black,scaling=consrained); The marix of coefficiens has wo real disinc negaive eigenvalues λ =, λ =. The equilibrium poin is a sable node.
28 43 Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions
Let ( α, β be the eigenvector associated with the eigenvalue λ i
ENGI 940 4.05 - Sabiliy Analysis (Linear) Page 4.5 Le ( α, be he eigenvecor associaed wih he eigenvalue λ i of he coefficien i i) marix A Le c, c be arbirary consans. a b c d Case of real, disinc, negaive
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationSolutions for homework 12
y Soluions for homework Secion Nonlinear sysems: The linearizaion of a nonlinear sysem Consider he sysem y y y y y (i) Skech he nullclines Use a disincive marking for each nullcline so hey can be disinguished
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationChapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis
Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationChapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis
Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More information(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)
Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationMath 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm
Mah Spring 6 Differenial Equaions Final Exam Due Dae: Tuesday, June 6, 5pm Your name (please prin): Insrucions: This is an open book, open noes exam. You are free o use a calculaor or compuer o check your
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationApplication 5.4 Defective Eigenvalues and Generalized Eigenvectors
Applicaion 5.4 Defecive Eigenvalues and Generalized Eigenvecors The goal of his applicaion is he soluion of he linear sysems like where he coefficien marix is he exoic 5-by-5 marix x = Ax, (1) 9 11 21
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationSignal and System (Chapter 3. Continuous-Time Systems)
Signal and Sysem (Chaper 3. Coninuous-Time Sysems) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 1 Dep. Elecronics and Informaion Eng. 1 Nodes, Branches, Loops A nework wih b
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationConcourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations
Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More information1 st order ODE Initial Condition
Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 1 1 s order ODE Iniial Condiion f, sandard form LINEAR NON-LINEAR,, p g differenial form M x dx N x d differenial form is equivalen o a pair of differenial
More informationMA 366 Review - Test # 1
MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationMA Study Guide #1
MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationSecond-Order Differential Equations
WWW Problems and Soluions 3.1 Chaper 3 Second-Order Differenial Equaions Secion 3.1 Springs: Linear and Nonlinear Models www m Problem 3. (NonlinearSprings). A bod of mass m is aached o a wall b means
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationMEI STRUCTURED MATHEMATICS 4758
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon
More informationCh.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec
More informationKEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations
ADVANCED MATHEMATICS FOR ECONOMICS - /3 Shee 3: Di erenial equaions Check ha x() =± p ln(c( + )), where C is a posiive consan, is soluion of he ODE x () = Solve he following di erenial equaions: (a) x
More informationThe equation to any straight line can be expressed in the form:
Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationMTH Feburary 2012 Final term PAPER SOLVED TODAY s Paper
MTH401 7 Feburary 01 Final erm PAPER SOLVED TODAY s Paper Toal Quesion: 5 Mcqz: 40 Subjecive quesion: 1 4 q of 5 marks 4 q of 3 marks 4 q of marks Guidelines: Prepare his file as I included all pas papers
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More information2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
More informationProblem set 6: Solutions Math 207A, Fall x 0 2 x
Problem se 6: Soluions Mah 7A, Fall 14 1 Skech phase planes of he following linear ssems: 4 a = ; 9 4 b = ; 9 1 c = ; 1 d = ; 4 e = ; f = 1 3 In each case, classif he equilibrium, =, as a saddle poin,
More informationElementary Differential Equations and Boundary Value Problems
Elemenar Differenial Equaions and Boundar Value Problems Boce. & DiPrima 9 h Ediion Chaper 1: Inroducion 1006003 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationMorning Time: 1 hour 30 minutes Additional materials (enclosed):
ADVANCED GCE 78/0 MATHEMATICS (MEI) Differenial Equaions THURSDAY JANUARY 008 Morning Time: hour 30 minues Addiional maerials (enclosed): None Addiional maerials (required): Answer Bookle (8 pages) Graph
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationWall. x(t) f(t) x(t = 0) = x 0, t=0. which describes the motion of the mass in absence of any external forcing.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 7 Mechanics applicaions of second-order ODEs Second-order linear ODEs wih consan coefficiens arise in many physical applicaions. One physical sysems whose behaviour
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationOutline of Topics. Analysis of ODE models with MATLAB. What will we learn from this lecture. Aim of analysis: Why such analysis matters?
of Topics wih MATLAB Shan He School for Compuaional Science Universi of Birmingham Module 6-3836: Compuaional Modelling wih MATLAB Wha will we learn from his lecure Aim of analsis: Aim of analsis. Some
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationFishing limits and the Logistic Equation. 1
Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationRobotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationReview - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y
Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More information