Math-303 Chapter 7 Linear systems of ODE November 16, Chapter 7. Systems of 1 st Order Linear Differential Equations.

Transcription

1 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 Chaper 7 Sysems of s Order Liear Differeial Equaios saddle poi λ >, λ <

2 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7

3 Mah-33 Chaper 7 Liear sysems of ODE November 6, LINER SYSTEMS OF THE s ORDER ODE s Liear sysem P + g ( ) iiial codiio (4) p + p + + p + g p + p + + p + g p + p + + p + g Homogeeous sysem P Reducio of h order liear ODE o a sysem of s order ODEs: ( )... y + ay + + ay g y y y 3 y 3 3 y 4 y ( ) y ( ) y ( ) y y a... a + g Soluio, parameric graph Modelig of iercoecig as (7. #) y, Eisece Theorems (7.. ad 7..)

4 Mah-33 Chaper 7 Liear sysems of ODE November 6, Review of Marices Mari m a a a a a a am am am ( aij ) m a a a a a a a a a ( aij ) square mari Vecor Traspose T ( a ji ) ( i ) colum vecor T (,,, ) Cojugae ( a ij ) ( ) i djoi T Self-adjoi (Hermiia) if * (for real marices, T symmeric) Mari lgebra: B a ij b for all i ad j ij Ι I a + B ( aij + b ij ) + B B+ + + ( a ij ) m B p ( cij ) m p c ab (i geeral, B B ) ij i j I I for square marices for square marices

5 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 5 Mari iverse I (if de, he iverse eiss) ( mari) a b d b c d ad bc c a Gaussia elimiaio I I row reducio Producs of vecors: T y y i i i ( y, ) y i i i T y ier (scalar) produc Properies: ( y, ) ( y, ) ( α, y ) α ( y, ) (,α y ) α ( y, ) ( y, + z ) ( y, ) + ( z, ) Norm (, ) Orhogoaliy y ( y ) if, 3-D coordiae vecors i, j, Mari Fucios a a a a a a am am am ( aij ), ( ij ) a ( d) d a ij

6 Mah-33 Chaper 7 Liear sysems of ODE November 6, Sysems of Liear lgebraic Equaios Sysem of algebraic equaios b ugmeed mari b RREF Soluio Liearly idepedece vecors,,..., are liearly idepede if c + c +...+c oly if all c vecors of legh :,,..., Fac: [ ] de... m m m m,,..., are liearly idepede Eigevalue problem: λ Solve characerisic equaio: λi λ are called eigevalues Fid eigevecors by solvig λ I is called a eigevecor correspodig o eigevalue λ ) Real disic eigevalues ( λ λ)( λ λ) ( λ λ) ) Roo of mulipliciy s s There eis liearly idepede λ eigevecors,,, correspodig o λ, λ,..., λ λ There ca be more ha oe li.idep.,, m correspodig o λ (m is called geomeric mulipliciy) (s is called algebraic mulipliciy) 3) Comple roos λ α + βi b + ib appear i cojugae pairs λ α βi b ib

7 Mah-33 Chaper 7 Liear sysems of ODE November 6, Basic Theory of Sysems of s Order Liear Differeial Equaios Mari-vecor oaios:, m m m m, P p p p p p p, p p p c c c c Homogeeous Sysem P ( a,b) (3) Iiial codiios Superposiio priciple: If, are soluios of (3), he c + c is also a soluio (Th 7.4.) Liear depedece I is said ha ( ), ( ),..., are liearly depede o a,b if here eiss a se of cosas c,c,...,c o all equal o zero, such ha. + for all ( a,b) c c +...+c. Oherwise, ( ), ( ),..., are liearly idepede o a,b. Wrosia W de ( ) ( )... Soluios ( ), ( ),..., are liearly idepede a, if W Theorem 7.4. If ( ), ( ),..., are liearly idepede soluios of P, he ay soluio of (3) ca be wrie as φ c + c ( ) +...+c Theorem If ( ), ( ),..., are soluios of (3) i a,b, he i ( a,b ) or W i W de... a,b. Theorem W ce p ( ) p d Eisece of a leas oe fudameal soluio Fudameal mari [... ] Ψ W de Ψ, ( a,b) Geeral soluio Ψc c + c ( ) +...+c Soluio of IVP P Ψ Ψ

8 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 8 Fudameal ses for homogeeous liear sysems wih cosa coefficies, where a a a a a a a a a, e λ, Trial form: 7.5 I Real disic eigevalues Characerisic equaio λ Fid eigevecors by solvig I ( λ λ)( λ λ) ( λ λ) I λ There eis liearly idepede eigevecors,,, correspodig o λ, λ,..., λ The fudameal se: e, e,, e λ λ λ s s 7.8 II Repeaed eigevalues Characerisic equaio λ Fid eigevecors by solvig I s λ λ roo of mulipliciy s λ I (algebraic mulipliciy) Case If here eis liearly idepede eigevecors,,, s correspodig o λ (geom.) λ λ λ The fud. se : e, e,, e s s Case If here eiss oly oe idepede eigevecor correspodig o λ The solve ( λ ) I p I q p λ To fid vecors p, q,... λ λ λ The fud. Se: e, e p, e p q, III Comple eigevalues Cojugae pair of comple roos λi λ α + βi λ α βi λ Fid eigevecors by solvig α The fudameal se: ( a β b β ) I a+ b a b e cos si ( β β ) a b α e si + cos i i

9 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 9 a + by a b Plae Sysem y c + dy c d, Characerisic Equaio: I Eigevalues: λ λ a + d λ+ ad bc a + d ± a + d 4 ad bc Tr ± λ, I >, λ λ R, c e λ λ + c e PHSE PORTRIT soluio r curve direcio vecor usable ode λ >, λ > sable ode λ <, λ < saddle poi λ >, λ < II, λ λ λ R a) Two idepede, b) Oe idepede (fid p ) λ λ c e + c e λ + ( + p ) c e c e λ proper usable ode λ > proper sable ode λ < improper usable ode λ > III < λ, α ± βi, a± b a), i α, ( a β b β ) + ( a β + b β ) c cos si c cos si e α usable spiral poi α > sable spiral poi α < b) α, λ, βi ±, c ( acos β bsi β) + c ( acos β + b si β) ceer

10 Mah-33 Chaper 7 Liear sysems of ODE November 6, Fudameal Mari Sysem of ODE s P, ( ) Geeral soluio: c + c +...+c Ψ P [... ] Ψ Ψ PΨ Ψ c Geeral soluio Ψ Ψ Soluio of IVP P Φ h e [... ] Φ Φ PΦ Φ I Φ I Φ c Φ Φ ( ) Ψ Ψ Φ e The mari epoeial fucio ( is a cosa mari): e I !!! e Φ ( e ) e e I Ψ Ψ ( Φ) Φ Φ e Φ ad I e are soluios of he same IVP e

11 Mah-33 Chaper 7 Liear sysems of ODE November 6, Soluio of he o-homogeeous sysem P + g ( ) I Diagoalizaio ) Solve Eigevalue Problem: λi λ, λ,..., λ,,,, ) Cosruc a rasformaio mari T [ ] 3) Fid iverse (if eigevalues are li.id.) T (Trasformaio mari diagoalizes ): T T D, 4) Calculae eries h i T g 5) Defie he ew variable Ty h i D λ λ λ Solve equaios for y,..., y : + y Dy T g (equaios are ucoupled) y c e + e e h d λ λ λ y c e + e e h d λ λ λ 6) Obai he geeral soluio by Ty II Variaio of parameer Fudameal mari Ψ [... ] Paricular soluio: Ψ Ψ g p d Geeral soluio: Ψ c Ψ Ψ g + d + Soluio of IVP wih he help of Ψ : Ψ Ψ Ψ Ψ g s s ds Soluio of IVP wih he help of Φ : Φ Φ Φ g + s s ds III Udeermied coefficies

12 Mah-33 Chaper 7 Liear sysems of ODE November 6, 7