8. Applications To Linear Differential Equations
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1 8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5. The Eistece-Uiqueess Theorem 8.6. The Dimesio Of The Solutio Space Of A Homogeeous Liear Differetial Equatio 8.7. The Algebra Of Costat-Coefficiet Operators 8.8. Determiatio Of A Basis Of Solutios For Liear Equatios With Costat Coefficiets By Factorizatio Of Operators 8.9. Eercises 8.0. The Relatio Betwee The Homogeeous Ad Nohomogeeous Equatios 8.. Determiatio Of A Particular Solutio Of The Nohomogeeous Equatio. The Method Of Variatio Of Parameters 8.. Nosigularity Of The Wrosia Matri Of N Idepedet Solutios Of A Homogeeous Liear Equatio 8.3. Special Methods For Determiig A Particular Solutio Of The Nohomogeeous Equatio. Reductio To A System Of First-Order Liear Equatios 8.4. The Aihilator Method For Determiig A Particular Solutio Of The Nohomogeeous Equatio 8.5. Eercises
2 8.. Itroductio Ay equatio ivolvig the derivatives, y, y,, of a fuctio y is called a differetial equatio (DE). Ay fuctio y f that satisfies the DE is called a solutio. The order of a DE is the order of the highest derivative i it. Eamples: (a) y y is st order with solutios y ce, where c is ay costat. (b) y 6y 0 is d order with solutios y c cos4 c si 4, where c ad c are arbitrary costats. Eperieces have show that it is hopeless to see methods for solvig all DE s. Istead, it is more fruitful to as whether a give DE has ay solutio, ad what properties of the solutios, if eist, ca be deduced from the DE. Thus, DEs provide a ew source of fuctios. Results of great geerality about the solutios of DEs are usually difficult to obtai. However, there are a few eceptios, e.g., the liear DEs that are our oly cocer.
3 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8... st Order LDE 8... d Order LDE
4 8... st Order LDE A liear differetial equatio (LDE) of st order taes the form y P y Q dy y (8.) d where P ad Q are ow fuctios ad y is the uow. This is oe of the few type of DEs that ca be solved by elemetary meas, i.e., through algebraic operatios, compositios, ad itegratios. Theorem 8.. Let P ad Q be cotiuous o a ope iterval J. The there eists ad oly fuctio y f that satisfies both (8.) ad the iitial coditio f a b, where a J ad b R. Furthermore, A A At f be e dt Q t e (8.) where A dt P t. a a Proof, we see that Usig A P A A A e y e A y e Q e A y A e Q A e 8. gives (8.a) A The factor e is called a itegratig factor sice we ca ow itegrate (8.a) over the iterval a, to get A Aa At a e f e f a dt Q t e a Usig Aa dt P t 0 ad f a a solutio is guarateed by Theorem 8.3 of 8.5. b, eq(8.) is recovered. Uiqueess of this
5 8... d Order LDE LDEs of the d order are of the form y P y P y R If the coefficiets P, P ad R are cotiuous over some ope iterval J, the a eistece theorem guaratees that solutios always eists over J [see 8.5]. However, there is o geeral formula aalogous to (8.) so that the theory is far from complete, ecept i special cases. Theorem 8.. Cosider the DE y ay by 0 (8.3) where a ad b are real costats. Every solutio of (8.3) o, has the form a / y e cu cu (8.4) where c ad c are costats ad the fuctios u ad u are determied accordig to the sig of the discrimiat (a) If 0 (b) If 0 (c) If 0 d a 4b as follows: d, the u ad u. d, the u e ad d, the u ad cos u e, where u si, where d. d. Commet By replacig each th order derivative y () by the th power moomial r, we tur a LDE with costat coefficiets ito a polyomial equatio, ow as the characteristic equatio of the LDE. For eq(8.3), this is r ar b 0 (8.5) with roots ad r a d r a d Thus, d is so-called because it is actually the discrimiat of (8.5) that determies the ature of its solutios.
6 Assumig we are worig with comple umbers, it is the easily show that all solutios of a LDE with costat coefficiets are liear combiatios of fuctios of the r form f e, where r is a root of the characteristic equatio of the LDE. Theorem 8. the follows by re-epressig everythig i terms of real fuctios usig i i cos e e i si
7 8.4. Liear Differetial Equatios Of Order N Discussios of LDEs are more easily carried i the operator otatios. Let C J be the liear space of all real-valued fuctios cotiuous o a iterval J, which ca be ubouded. Let C J be the subspace cosistig of all real fuctios whose first derivatives are also cotiuous o J. Let P,, P be give fuctios i C J. We defie a operator L as the trasformatio by L : C J C J f L f f P f P f 0 where P0 ad f 0 f. Itroducig the derivatio operators D, we have where L D P D 0 D. L y P D 0 A LDE of order has the form R (8.6) where R is some ow fuctio o J. A solutio of the LDE is ay fuctio i C J that satisfies (8.6) o J. Usig the liearity of D, it is easy to show that every D, ad hece L, are also liear. Hece, L defied above is called a differetial operator of order. With each ohomogeous equatio L y R is associated a homogeeous equatio L y 0. Let h be the most geeral solutio of the homogeeous equatio ad p be ay particular solutio of the ohomogeeous equatio. Usig the liearity of L, it is easily show that the most geeral solutio of the ohomogeeous equatio is give by h p. The set of all solutios of the homogeeous equatio is the ull space N L, also called
8 the solutio space of L y 0. The solutio space is a subspace of C J. Although dim C J may be ifiite, dim N L is always fiite. I fact, dim N L, the order of L [see 8.5].
9 8.5. The Eistece-Uiqueess Theorem Theorem 8.3: Eistece-Uiqueess Theorem. Let P,, P be cotiuous fuctios o a ope iterval J ad let L be the liear differetial operator L D P D P D 0 where 0 D ad P0. Let 0 J ad let 0,,, be give real umbers. The there eists ad oly fuctio y f that satisfies both the homogeeous equatio L y 0 o J ad the iitial coditios j f 0 j for j 0,,, Proof Proof of this will be obtaied as a corollary of a more geeral theorem for systems of differetial equatios to be discussed i Chapter 0. A specialized versio for LDEs with costat coefficiets is give i 9.9. Commet The iitial coditios ca be represeted by a vector i R with compoets f, f,, f This is also called the iitial- value vector of f at 0. The uiqueess theorem the implies oly f 0 ca satisfy a iitial-value vector that equals to O.
10 8.6. The Dimesio Of The Solutio Space Of A Homogeeous Liear Differetial Equatio Theorem 8.4: Dimesioality Theorem. Let L : C J C J be a liear differetial operator of order give by L D P D (8.7) The the solutio space of the homogeeous equatio L y 0 has dimesio. Proof Let T : N L R by f T f f, f,, f be the liear trasformatio that maps each fuctio f i the solutio space N L to the iitial- value vector of f at 0 J. By the uiqueess theorem, T f O implies f 0. Hece, by Theorem 4.0, T is - o N L so that T eists ad is also -. Furthermore, sice every elemet i R ca be a iitial-value vector, the map is oto. By Theorem 4., Theorem 8.5. dim N L R. Let L : C J C J be a liear differetial operator of order. Let u,, u be idepedet solutios of the homogeeous equatio L y 0 o J. The every solutio y f o J ca be epressed as a liear combiatio f c u (8.8) where c are costats.
11 Proof This is a corollary of Theorem 8.4. Commet Eq(8.8) is called a geeral solutio of L y 0. The dimesioality theorem asserts that a th order LDE always has idepedet solutios. There is however o geeral method for obtaiig these solutios, ecept for a few special types of equatios, e.g., those with costat coefficiets.
12 8.7. The Algebra Of Costat-Coefficiet Operators A costat- coefficiet operator A is a liear operator of the form A ad 0 a D a D a D a (8.9) 0 where D is the derivative operator ad a are real costats. If a0 0, the A is of order. We ca apply A to a fuctio y defied o some iterval J to obtai aother fuctio o J give by A y 0 a y I this sectio, we shall cosider oly fuctios the derivative of which eists for every order o the iterval,. The set of all such fuctios is deoted by C ad is called the class of ifiitely differetiable fuctios. If y C, so is A y. It is easy to show that a costat coefficiet operator is also a liear trasformatio, as defied i 4.. Furthermore, ay such operators A ad B commute, i.e., AB BA. Give A i (8.9), its associated polyomial is defied as pa r ar 0 (8.9a) Coversely, give ay polyomial of real coefficiets, there is a correspodig operator A with the same costat coefficiets. Theorem 8.6. Let A ad B be costat coefficiet operators with associated polyomials p A ad p B, respectively. Let be ay real umber, the (a) A B iff pa pb (b) pab pa pb p p p (c) AB A B (d) p A pa I other words, the associatio betwee operators ad polyomials is - oto ad they satisfy the same algebraic relatios. Proof Parts (b)-(d) follow immediately from defitio (8.9a). Hece, we shall prove oly (a).
13 Let pa pb, the they must have the same degree as well as coefficiets. This meas A ad B have the same order ad coefficiets. Hece, A B. This proves the if part of (a). Net, give A B, we have A y B y y C. Let y e r, where r is a costat. The y r r e r y 0. Hece, r B y p re r A y p r e A B r Sice e 0, we have pa pb, which proves the oly if part of (a). Commet Accordig to Theorem 8.6, costat coefficiet operators ca be maipulated lie polyomials. For eample, if p A (r) ca be factorized as A so ca A: 0 (8.0) p r a r r A a D r 0 (8.0a) [Note that the factors o the right sides commute.] Now, the fudametal theorem of algebra guaratees the factorizatio (8.0). Furthermore, if the coefficiets of p A (r) are real, the the roots r must either be real or else occur i cojugate pairs: i. I the latter case, we have r i r i r r Thus, p A (r) ca always be factorized ito a product of liear ad quadratic factors of real coefficiets. Ditto A. Eample. Let A D D 5 6. The 3 A D D 5 6 r r 3 ad pa r r
14 Eample. Let ad 4 3 A D D D D. The 4 3 p r r r r r A r r A D D
15 8.8. Determiatio Of A Basis Of Solutios For Liear Equatios With Costat Coefficiets By Factorizatio Of Operators Theorem Case I: Real Distict Roots Case II: Real Roots, Some Repeated Case III: Comple Roots
16 8.8.. Theorem 8.7. Let L be a costat coefficiet operator that ca be factorized as a product of costat coefficiet operators L A j j The the solutio space of L y 0 cotais the solutio space of each 0 i orther words, Proof j N L N A j,, (8.) A y. Let u N A i. Sice the Aj factors commute, we ca move Ai to the rightmost j positio so that Lu Aj Ai u O. Hece, u N A i u N L QED. ji j. Commet If Lu O, the L is said to aihilate u. Thus, Theorem 8.7 ca be rephrased as follows. If a factor A j of L aihilates u, the L aihilates u.
17 8.8.. Case I: Real Distict Roots Eample. Fid a basis of solutios for the differetial equatio Solutio 3 D D y (8.) Eq(8.) ca be writte as L y 0 with 7 6 D D D 3 3 L D D Now, D u 0 u e i.e., the ull space of D is N D e 3 Similarly, N D e ad N D 3 e idepedet, the geeral solutio for (8.) is y c e c e c e 3 3, where deotes the liear spa.. Sice these fuctios are Theorem 8.8. Let L be a costat coefficiet operator whose associated polyomial p L (r) has distict roots r,, r. The the geeral solutio of L y 0 o, is y r ce (8.3) Proof By (8.0), we have Sice L a D r 0
18 D r u 0 u e so that the ull space N D r r e r. Sice the e r s are idepedet, we have r ;,, N L e QED.
19 Case II: Real Roots, Some Repeated If all roots are real but ot distict, the the u s i (8.4) are ot idepedet so that the geeral solutio (8.3) eeds to be accordigly modified. Theorem 8.9. The m fuctios r with,, u e are m idepedet elemets aihilated by D r m. m Proof Idepedece of the u s follows from that of the moomials. The proof that each u is aihilated by D r m is by iductio o m. The case m is trivial. Thus, assumig case m holds, we wish to prove case m. To be more specific, we assume the fuctios u,, um Hece, are all aihilated by D r m r D rum D r e m. Now, m r m e m u m m m D r u m D r u 0 m m where the iductio hypothesis was used i the last equality. QED. Eample. Fid the geeral solutio of L y 0, where Solutio Hece, 3 L D D D 8 3 L D D
20 3 e, e ad N D 3 e N D so that the geeral solutio is y c e c e c e 3 3 Eample Solve D D D D y 0. Solutio From we get D D D D D D D y c c c3e c4 c5 c6 e
21 Case III: Comple Roots If comple epoetials are used, there is o eed to distiguish betwee real ad comple roots. If real-valued solutios are desired, each pair of cojugate roots i should be combied ito a quadratic factor Q D D (8.5) whose ull space is N Q e cos, e si simple geeralizatio of Theorem 8.9 gives m cos, si ;,, N Q e e m Eample 4. y 4 y 3y 0 The associated polyomial is solutio is y c e c cos3 c si r 4r 3r. I case of multiplicity m, a, with roots 0, ad 3i. The geeral Eample 5. y y 4 y 8y 0 The associated polyomial is r 3 r 4r 8 r r 4 The geeral solutio is y c e c cos c si 3, with roots, ad i. Eample y 9 y 34 y 66y 65y 5y 0 The associated polyomial is geeral solutio is r r 4r 5, with roots, i ad i. The cos y ce e c c3 c4 c5 si
22 8.0. The Relatio Betwee The Homogeeous Ad Nohomogeeous Equatios Theorem 8.0. Let L : C J C J be a liear differetial operator of order. Let u,, u be idepedet solutio of the homogeeous equatio L y 0. Let y be a particular solutio of the ohomogeeous equatio L y R, where R C J. The the geeral solutio for L y R is where Proof y f y c u c (8.6) are costats. Let f be ay solutio of L y R. Sice L is liear, we have L f y L f L y R R 0 Hece, f y N L so that f y c u. QED. Commet Thus, Theorem 8.0 breas dow the problem of solvig a ohomogeeous equatio ito parts. Oe is to fid a particular solutio, the other is to solve the homogeeous equatio. This has a simple geometric aalogy: to fid all poits o a plae, we fid a particular poit o the plae, the add to it all poits o the parallel plae that goes through the origi.
23 8.. Determiatio Of A Particular Solutio Of The Nohomogeeous Equatio. The Method Of Variatio Of Parameters 8... Method Of Variatio Of Parameters 8... Eamples
24 8... Method Of Variatio Of Parameters Give idepedet solutios u,, u of the th order homogeeous equatio L y 0, the method of variatio of parameters provides a particular solutio to the ohomogeeous equatio L y R of the form y u (8.7) where are fuctios to be determied. I terms of vector otatios, (8.7) ca be writte as a ier product y, u u (8.8) where v v ad u u u,,,, The derivative of the vector- valued fuctio is defied to be v,, v The Leibiz rule fg f g f g also applies to the ier product. Thus, y, u, u (8.9) The s are to be determied from the followig coditios,, u 0 for 0,,, where (8.9a), u R (8.9b) 0 u u. The meaig of these coditios are as follows: Usig (8.9a), the st derivatives of y become,u y, u, u
25 ,u y, u, u y, u, u, u With the adiitioal help of (8.9b), the th derivative becomes y, u, u R, u Thus, if where L D P D 0 D, we have L y y P y R, L u R R, u P, u where we have used Lu O. Therefore, the coditios (8.9a,b) will mae y of (8.7) a particular solutio of the ohomogeeous equatio. With the help of the Wrosia matri of u,, u, i.e., W u u u u u u u u u eqs(8.9a,b) ca be throw ito the matri form W V R E where (8.0) 0 where V is the colum matri versio of ad E v 0 coordiate vector. As a chec, the ith compoet of (8.0) is i WiV u, u i R i is that of the th uit
26 I 8., we shall show that W is osigular. Hece, V R W E (8.0a) W eists ad (8.0) becomes Itegratig over a iterval c, Hece, J, we have Z V V c dt R t W t E V V c Z so that (8.8) becomes y c u v u, c z u, c u, z, where z is the vector whose colum matri form is Z. Now, u, c is a liear combiatio of the u s with costat coefficiets c. Therefore,, L u c O. Thus, the particular solutio is give by y u z where, Thus, we have proved the followig theorem. Theorem 8.. Z dt R t W t E c Give idepedet solutios u,, u of the th order homogeeous equatio L y 0 L y R o a iterval J. The a particular solutio to the ohomogeeous equatio is y u where the s are the compoets of the colum matri V give by
27 V dt R t W t E where c, J. c
28 8... Eample Fid the geeral solutio of y y e o the iterval,. Solutio The idepedet solutios of the homogeous equatio D ad u e The Wroia matri of u ad u is u e y 0 are W e e e e with iverse W e e e e Hece, 0 e W E W e t e R tw t E t t e e so that t e dt e t e e 0 l l t e dt t e e 0 l l Sice the costat terms ca be absorbed ito the homogeeous solutios, we have
29 y e l e u l e u e e e l e so that the geeral solutio is y e e e l e c e c e
30 8.. Nosigularity Of The Wrosia Matri Of N Idepedet Solutios Of A Homogeeous Liear Equatio Cosider the Wrosia matri W of idepedet solutios u,, u homogeeous equatio L y y P y 0 o a iterval J. Let w detw (8.3a). The we have: of the Theorem 8.. w P w (8.3) 0 o J. Hece, c J, we have w wcep dt P t [ Abel s formula ] (8.4) c Furthermore, w 0 J ad W is osigular o J. Proof Let u be the row vector u u u w,,. Hece, we ca write detw det u, u,, u, u Accordig to E.7, 5., we have w det u, u,, u, u O the other had, Hece, det,,,, P w u u u P u det,,,, w P w u u u u P u Sice u are solutios of (8.3a), the rows of the determiat are depedet. This proves (8.3). The Abel s formula follows by itegratig (8.3) over c,.
31 Sice e 0 for all real, we see that w 0 for all if there eists a c J such that wc 0. The eistece of c ca be show by a cotradictio argumet. Thus, assume that wt 0 t J. We the choose a fied poit t t0 ad cosider the system of liear equatios 0 W t X O (a) where X is a colum vector. Sice wt W t solutio, say, f t cu t det 0, there eists a ozero 0 0,, t X c c O. Cosider the the liear combiatio Obviously, L f 0 sice 0 L u for all. Also, puttig X ito eq(a) gives i W t c u t c 0 i 0 i t f for i,, 0 0 which meas the iitial-value vector of f at t t0 is O. By the uiqueess theorem, f t O for all t, which meas c 0 for all, cotrary to our origial assumptio. QED.
32 8.3. Special Methods For Determiig A Particular Solutio Of The Nohomogeeous Equatio. Reductio To A System Of First-Order Liear Equatios Although the method variatio of parameters ca always provide a particular solutio to the ohomogeeous equatio, it may ot be the simplest way to do so i specific cases. Eample Fid a particular solutio of D D y e (8.5) Solutio Let u D y. The (8.5) becomes D u e which is a st order equatio that ca be solved usig Theorem 8. or by ispectio. This gives so that u e D y u e Usig Theorem 8., we have, with y 0 0, t t y e dt e 0 Eve though the itegral caot be evaluated i terms of elemetary fuctios, we ca cosider the problem solved. The geeral solutio is t t 0 y c e c e e dt e
33 8.4. The Aihilator Method For Determiig A Particular Solutio Of The Nohomogeeous Equatio The Aihilator Method Eample Eample Commet
34 8.4.. The Aihilator Method The aihilator method ca be applied to fid a particular solutio for the equatio L y R if. L is a costat coefficiet operator.. R is aihilated by aother costat coefficiet operator A, i.e., AR 0. The priiciple of the method is simple. Applyig A to L y R gives AL y 0, which must be satisfied for all y that satisfy L y R. Sice AL is also a costat coefficiet operator, we ca determie its ull space N AL via its associated polyomial. The problem the reduces to selectig oe y N AL that also satisfies L y R.
35 8.4.. Eample Fid a particular solutio of Soultio 4 4 D y 6 (a) The right-had side, beig a polyomial of degree 4, ca be aihilated by the operator 5 D. Hece, a solutio y of (a) also satisfies 5 4 D D y 6 0 (8.6) The roots of the associated polyomials are 0, 0, 0, 0, 0,, ad i. Hece, y c c c c c c e c e c cos c si (b) Our tas is to fid a set of c s so that 4 4 L y D y 6 Sice the last 4 terms i (b) are aihilated by L, we ca set c6 c7 c8 c9 0. Furthermore, to avoid worig with subscripts, we set so that 4 3 y a b c d e 4 D y a a 4! L y a a b c d e 4 Sice this holds for all, the coefficiet of each power of must vaish. Hece, a b c 0 6 so that the desired particular solutio is y d e a 5 6
36 Eample Solve y 5y 6y e. Solutio We start by rewritig the equatio as 5 6 L y D D y R e The homogeeous equatio L y D D 3 y 0 has idepedet solutios 3 ad u e u e For the particular solutio y, we otice that R e is aihilated by A D. Hece, y must satisfies D D D 3 y 0 whose geeral solutio is y ae be ce de 3 Sice L aihliates the last terms, we ca write so that y ae be ad Dy a b e be D y a b e be L y a b 5 a b 6a e b 5b 6b e a 3b e be R e a 3b 0 ad b Hece b ad The geeral solutio is therefore 3 3 a b 4
37 3 y c e c e e e 4 3
38 Commet The aihilator method wors wheever we ca fid a costat coefficiet opeator A to aihilate the ohomogeeous term R. For coveiece, we list below the aihilators for some commoly ecoutered fuctios. m e Fuctio m Aihilator D m e D m e D m cos or si cos or cos or e si m e cos or D m si D m e si D m D D D m
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