Chater 3. Higher Order Liear ODEs Kreyszig by YHLee;4; 3-3. Hmgeeus Liear ODEs The stadard frm f the th rder liear ODE ( ) ( ) = : hmgeeus if r( ) = y y y y r Hmgeeus Liear ODE: Suersiti Pricile, Geeral Sluti Therem Suersiti ricile Fr the hmgeeus liear ODE, sums ad cstat multiles f slutis are als slutis. A geeral sluti f the hmgeeus liear ODE is f the frm = y c y c y A articular sluti is btaied whe cc,, c are secified fuctis, y ( ) y ( ) y ( ), if the equati ky ky ky = is satisfied ly fr k = k = = k =., are liearly ideedet Otherwise, they are liearly deedet, ad e is eressed by a liear cmbiati f the thers y = ( ky ky) k Eamle Liear Deedece Shw y =, y = 5, y = are liearly deedet ay iterval. y = y.5y 3 Eamle Liear Deedece Shw y =, y =, y = are liearly ideedet. Make k k k = Try three its =, =, = k k k3 =, k k k3 =, k 4k 8k3 = It ca be shw frm these equatis k = k = k = But a mre strict methd is eeded ad discussed later.
Eamle 3 Geeral sluti, basis Slve the furth rder ODE IV y 5y 4y = Kreyszig by YHLee;4; 3- Try y = e λ ad btai the characteristic equati 4 λ 5λ 4= λ=±, ± The geeral sluti y= c e c e c e c e 4 Iitial Value Prblem. Eistece ad Uiqueess A iitial value rblem has iitial cditis ( y( ) = K, y ( ) = K, y ) ( ) = K, Therem Eistece ad uiqueess If the cefficiets ( ), ( ) are ctiuus sme e iterval I ad is i I, the the iitial value rblem has a uique sluti I. Eamle 4 A third rder Euler Cauchy equati Slve 3 y 3y 6y 6y y =, y =, y = 4 =, Try y= m 6 6= m =,, 3 3 m m m Sice,, are liearly ideedet, a geeral sluti is y= c c c Frm the iitial cditis y = c c c = y = c c 3c = y = c 6c = 4 c c c3 =, =, = The aswer is y= Liear Deedece f Sluti. Wrskia Wrskia f slutis is defied as the th rder determiat y y y y y y W( y,, y ) = ( ) ( ) ( ) y y y
Kreyszig by YHLee;4; 3-3 Therem 3 Liear deedece ad ideedece Whe the cefficiets ( ),, ( ), y are ctiuus sme e iterval I, the slutis y, are liearly deedet if ad ly if their Wrskia is zer fr sme = i I. W= i etire I if W= fr = The slutis are liearly ideedet if there is a I at which W. Prf: (a) Let y, be liearly deedet slutis., y The, the fllwig equati is satisfied with k, ky ky ky = that are t all zer., k The derivatives ky' ky ' ky ' =... ky ky ky = These equatis frm a hmgeeus liear system f equati. A trivial sluti f k,, k is btaied if the Wrskia is zer every I. (Cramer s therem) (b) If W= at i I, the fllwig equatis has a sluti k,, k that are t all zer. ( ) ( ) ( ) ky ky ky = ky' ky ' ky ' =... ( ) ( ) ( ) ky ky ky = We defie a sluti y = k y k y k y Frm the abve system f equatis, the iitial cditi is give as y = y' =... = y = Ather sluti, y =, eists, which satisfy the hmgeeus ODE ad the abve iitial cditis. Frm the uiqueess therem y = y y k y ( ) k y ( ) k y ( ) = = y,, y are liearly deedet (c) If W= at a i I, the slutis are liearly deedet i I by (b). If the slutis are liearly deedet, W= at all I by (a). If W Ad y, i I, W i all I. y are liearly ideedet i I.,
Kreyszig by YHLee;4; 3-4 A Geeral Sluti Icludes All Slutis Therem 4 Eistece f a geeral sluti If the cefficiets ( ),, ( ) are ctiuus sme e iterval I, the higher rder hmgeeus ODE has a geeral sluti I. Prf: By Therem, a uique sluti y() eists. Let y = y y y ad ( ) = ( ) = ( ) ( ) y y y' = y ' = y" = y3 " =... ( ) ( y ( ) ) = y ( ) = The Wrskia at y( ) y( )... y( )... y' ( ) y' ( )... y' ( )... =............ y y... y... y,, y are liearly ideedet. The liear cmbiati f y,, y = y c y c y c y = is a sluti f hmgeeus ODE. Therem 5 Geeral sluti icludes all slutis If the cefficiets ( ),, ( ) every sluti y Y( ) Y( ) = C y ( ) C y ( ), are ctiuus sme e iterval I, = is f the frm Prf: We will rve that Y ca be derived frm the geeral sluti y = cy cy cy At = ( ) c ( )... c ( ) = ( ) ' c '... c ' = ' cy y y Y cy y y Y......... ( ) ( ) ( ) ( c... c = ) cy y y Y This is a liear system f equatis i the ukws c,, c
y,, y are liearly ideedet. Wrskia f y, y is t zer., Ntrivial slutis f c,, c eist. (Cramer s rule) c,..., = C c = C Kreyszig by YHLee;4; 3-5 A articular sluti y = C y C y C y y ad Y satisfy the same iitial cditis. y( ) = Y( ) I by the Uiqueess Therem. This therem tells that there is sigular sluti, which cat be btaied frm a geeral sluti 3. Hmgeeus Liear ODEs with Cstat Cefficiets The stadard frm f th rder hmgeeus liear ODE with cstat cefficiets ( ) ( ) = y a y a y a y Try y = e λ ad btai the characteristic equati λ a λ a λ a = Slutis ca be give by umerical methd, r guess. Distict Real Rts All rts, λ,..., λ are real ad differet. The geeral sluti is y c e λ λ = c e The Wrskia f the slutis e e... e λ λ λ λ λ λ e e... e λ λ λ W = = e............ λ e λ e... λ e λ λ λ ( λ λ ) where V is the rduct f all factrs λj λ k with j < k. Eamle, fr =3, V = ( λ λ)( λ λ3)( λ λ 3).... λ λ... λ............ λ λ... λ ( ) / = V The Wrskia is t zer if ad ly if all the rts are differet. Therem Basis λ λ Slutis y = e,, y = e frm a basis if ad ly if all rts f the characteristic equati are differet. Therem Liear ideedece λ λm Ay umber f slutis y = e,, ym = e are liearly ideedet if ad ly if λ,, λ m are all differet.
Kreyszig by YHLee;4; 3-6 Simle Cmle Rts Cmle rts aear ly i a cjugate air sice the cefficiets f the ODE are real. λ=γ± iω The crresdig slutis are y e γ γ = cs ω, y = e siω Multile Real Rts If λ is a real rt f rder m, liearly ideedet slutis are λ λ m λ e, e,, e Prf : The left side f ODE is Ly = y a y ay ay [ ] Trial sluti y = e λ m L e = λ a λ a e = ( λλ) ( λλm ) ( λλ) e λ λ λ h( λ) λ is a rt f rder m ad λ,, λ are differet rts. m Differetiate bth sides with resect t λ. λ m λ m λ L e = m( λλ) h( λ ) e ( λλ) h( λ) e (9) Differetiati with resect t ad λ are ideedet. We rewrite the left term f Eq. (9) as λ λ λ L e L e L e Sice the right term i Eq. (9) is zer fr λ =λ L e λ = e λ is a sluti f the hmgeeus ODE Similarly, λ m λ m λ L e = m ( m)( λλ) h( λ ) e m( λλ) h( λ) e m λ m λ m( λλ) h( λ ) e ( λλ) h ( λ) e L e λ λ =, e is ather sluti. Reeat this util we btai the sluti f m e λ It ca be shw that the Wrskia f these slutis is zer ad therefre the slutis are ideedet.
Multile Cmle Rts If λ=γ iω is a cmle duble rt, s is The liearly ideedet slutis are γ γ γ γ e cs ω, e si ω, e cs ω, e siω λ =γiω. Kreyszig by YHLee;4; 3-7 Eamle Distict real rts Slve y y y y = The characteristic equati 3 λ λ λ = λ=,, The geeral sluti y= c e c e c e Eamle Simle cmle rts Slve y y y y =, with y y y The characteristic equati 3 λ λ λ = λ=, ± i The geeral sluti y = ce Acs Bsi Iitial cditis y' = c e Asi Bcs y" = c e Acs Bsi = 4, =, = 99. y = c A= 4, y = c B=, y = c A= 99 y = e 3cs si Eamle 3 Real duble ad trile rts Slve V IV y 3y 3y y = The characteristic equati 5 4 3 λ 3λ 3λ λ = λ =λ =, λ =λ =λ = 4 5 = ( 3 4 5 ) y c c c c c e Eamle 4 Multile Cmle Rts ( 7) ( 5) Slve y 8y 8y = 7 5 3 The characteristic eq. λ 8λ 8λ = λ =λ =λ =, λ =λ =3, i λ =λ = 3i 4 5 6 7 The geeral sluti is y= c c c A cs3 B si3 A cs3 B si3
3.3 Nhmgeeus Liear ODEs Kreyszig by YHLee;4; 3-8 The stadard frm is ( ) ( ) y y y y = r () Its crresdig hmgeeus equati ( ) ( ) y y y y = () The geeral sluti y = y y h Iitial Value Prblem csists f iitial cditis ( ) = = = y K, y( ) K,, y ( ) K Methd f Udetermied Cefficiets ODE with cstat cefficiets ( ) ( ) y a y a y a y= r( ) Rules fr the Methd f Udetermied Cefficiets fr y (A) Basic Rule Term i r ke γ k (=,,..) Chice fr y Ce γ K K.. K K kcsω Kcsω Msiω ksiω a ke csω a a e ( Kcsω Msiω ) ke siω (B) Mdificati Rule If the chice fr y is equal t the ther sluti f the ODE, k the try y = y with the smallest sitive iteger value f k. (C) Sum Rule r is a sum f the fuctis i the first clum, If try the sum f the crresdig chice fuctis.
Eamle Iitial value rblem. Mdificati rule Slve y 3y 3y y 3e y = 3, y = 3, y = 47 =, with Kreyszig by YHLee;4; 3-9 The characteristic equati 3 3 λ 3λ 3λ = λ = The trile rt λ = The geeral sluti f hmgeeus ODE y = ( c c c ) e h If we try y = Ce C 3C 3C C = 3, which has sluti. Try agai by the Mdificati Rule 3 y = C e The Isert these it ODE C=5 The geeral sluti y= c e c e c e e Iitial cditis y = c = 3 5 The sluti is (3 5 ) y e = 5 e Eamle Slve IV y y = 4.5e The characteristic equati 4 λ = λ=±, ± i The geeral sluti f hmgeeus ODE i i yh ce ce = c3e c4e y = ce ce Acs Bsi h We try y = Ce i the equati ad btai C=.3. Therefre y= ce ce Acs Bsi.3e
Methd f Variati f Parameters Kreyszig by YHLee;4; 3- It is used t btai y f the hmgeeus liear ODE whe ( ),, ( ) ( ) ( y y ) ( ) y ( ) y = r( ) ad r() are ctiuus sme e iterval I. It is give by W W y y r d y r d W = W( ) where y,, y is a basis f slutis, W is their Wrskia, ad W j is btaied by relacig the j th clum f W by the clum [... ]. Eamle 3 Variati f arameters Slve the hmgeeus Euler Cauchy equati. 3 4 y 3y 6y 6y= l, (>) Try m y= mm ( )( m) 3mm ( ) 6m 6= m =,, 3 y =, y =, y = The geeral sluti f hmgeeus ODE y = c c c h Calculate determiats W = =, 6 W = = 6 4 W = 3 = 3 6 3 W = = The articular sluti 4 3 y = ( l d ) ( l ) d ( l ) d 3 3 3 4 y = l 6 6 The geeral sluti 4 y = c c c3 l 6 6