Engineering Analysis ( & ) Lec(7) CH 2 Higher Order Linear ODEs
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1 Philadelphia Uiversit/Facult of Egieerig Commuicatio ad Electroics Egieerig Egieerig Aalsis (6500 & 6300) Higher Order Liear ODEs Istructor: Eg. Nada khatib
2 Higher order Liear ODE I this lecture we exted the cocepts ad methods of liear ODE from order = to arbitrar order This will be straightforward ad o ew ideas Higher Order Liear ODEs Eg. Nada khatib
3 Higher order Liear ODE However, the roots of characteristic equatio becomes larger with icreasig ad The Wroskia plas a more promiet rule Higher Order Liear ODEs Eg. Nada khatib 3
4 Higher order Liear ODE A th-order liear ODE is a ODE that ca be writte as: a (... a ( a0( b( Or as p (... p( p0( r( We call this the Stadard Form Higher Order Liear ODEs Eg. Nada khatib 4
5 Higher order Liear ODE p (... p( p0( r( Homogeous No Homogeous r( 0 r( 0 Higher Order Liear ODEs Eg. Nada khatib 5
6 Homogeeous Higher order ODE A geeral solutio of HODE of order is a solutio of the form: c ( c (... c ( G.S. form Where (, (,, (x ) is a basis of solutios of HODE of order, that is, these solutios are liearl idepedet Higher Order Liear ODEs Eg. Nada khatib 6
7 Homogeeous Higher order ODE The solutios (, (,, ( are liearl idepedet if ad ol if their Wroskia W is o-zero at some poit x 0 cotaied i ope iterval I 0... W Eg. Nada khatib 7 Higher Order Liear ODEs
8 Homogeeous Higher order ODE c ( c (... c ( G.S. form A particular solutio of HODE is obtaied if we assig specific values to the costats c, c,.c i G.S A iitial value problem for the ODE cosists iitial coditios d dx d dx ( x0 ) K0 ( x0) K ( x 0) K Higher Order Liear ODEs Eg. Nada khatib 8
9 Examples Higher Order Liear ODEs Eg. Nada khatib 9
10 Higher Order Costat Coefficiet Equatios Cosider the th-order homogeeous liear ODEs with costat coefficiets: a a ' a o 0 This ca be solved i a similar maer to the secod order case x Substitutig e Higher Order Liear ODEs Eg. Nada khatib 0
11 Higher Order Costat Coefficiet Equatios We obtai the characteristic equatio: a a a o The solutio will takes differet forms depedig whether the roots of this equatio are real distict roots, real multiple roots, complex distict roots or complex repeated roots. 0 Higher Order Liear ODEs Eg. Nada khatib
12 Higher Order Costat Coefficiet Equatios Case Real Distict Roots If all the roots are real ad differet, the the correspodig geeral solutio is: x c e... c e x Higher Order Liear ODEs Eg. Nada khatib
13 Higher Order Costat Coefficiet Equatios Case Multiple Real Roots If is a real root of order, the correspodig liearl idepedet solutios are: e x x x, xe, x e,..., x e x G.S. ( c c x... c x ) e x Higher Order Liear ODEs Eg. Nada khatib 3
14 Higher Order Costat Coefficiet Equatios Case 3 Complex Cojugate Roots e x e ( c / x cos ( c w x cos c w si ( / ) x w... c si w ( / ) Higher Order Liear ODEs Eg. Nada khatib 4
15 Higher Order Costat Coefficiet Equatios Case 4 3? Higher Order Liear ODEs Eg. Nada khatib 5
16 Higher Order Costat Coefficiet Equatios Case ( ) ( 3 4)? Higher Order Liear ODEs Eg. Nada khatib 6
17 Higher Order Costat Coefficiet Equatios Case 4, jw 3,? Higher Order Liear ODEs Eg. Nada khatib 7
18 Higher Order Costat Coefficiet Equatios Case 4 3, 4 jw? Higher Order Liear ODEs Eg. Nada khatib 8
19 Higher Order Costat Coefficiet Equatios Case 4,,3,4 jw? Higher Order Liear ODEs Eg. Nada khatib 9
20 Examples Higher Order Liear ODEs Eg. Nada khatib 0
21 Higher Order Nohomogeeous ODE Stadard Form: The geeral solutio of it is a solutio of the form p Where (... p( p0( r( h p c c (... c ( ) is a h ( x geeral solutio of the homogeeous ODE Higher Order Liear ODEs Eg. Nada khatib
22 Higher Order Nohomogeeous ODE Where ( ) p x is a particular solutio cotaiig o arbitrar costats Udetermied Coefficiets Methods Variatio of Parameters Higher Order Liear ODEs Eg. Nada khatib
23 Method of Udetermied Coefficiets The method of udetermied coefficiets applies whe ODE has costat coefficiets a -,. a 0 ad a a ' a o r( is a polomial fuctio, a expoetial, sie or cosie fuctio, or sums or products of such fuctios. 0 Higher Order Liear ODEs Eg. Nada khatib 3
24 Method of Udetermied Coefficiets p Ae x Acos wx Bsi wx k cos wx r( ke x, k si wx A e x x ( Acos wx Bsi w A x... A x A0 k e x cos kx wx, k e x si wx, 0,,... e e x x cos wx( A si wx( B x x... A... B 0 0 ) ) si wx, 0,.. Table. Method of Udetermied Coefficiets kx e x Higher Order Liear ODEs Eg. Nada khatib 4
25 Method of Udetermied Coefficiets Choice Rules : (a) Basic Rule. (b) Sum Rule. (c) Modificatio Rule. If a term i our choice for p happes to be a solutio of the homogeeous ODE, the multipl our choice of p b x k (where k is the smallest positive iteger such that o term of x k p is a solutio of the homogeeous ODE). Higher Order Liear ODEs Eg. Nada khatib 5
26 Examples Higher Order Liear ODEs Eg. Nada khatib 6
27 Method of variatio of parameters More geeral but more complex This method applies for DE with costat coefficiets ad Euler Cauch equatio ad With all r( Higher Order Liear ODEs Eg. Nada khatib 7
28 Method of variatio of parameters Eg. Nada khatib 8 r dx w w r dx w w r dx w w p... w w 0 0 w w Higher Order Liear ODEs
29 Examples Higher Order Liear ODEs Eg. Nada khatib 9
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