Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator

Transcription

1 Avances in Applie Mathematics, Article ID aama , available online at http: on Similar Operators an a Functional Calculus for the First-Orer Linear Differential Operator Davi Elizarraraz Departamento e Ciencias Basicas, Uni ersia Autonoma Metropolitana, Azcapotzalco, Mexico an Luis Vere-Star Departamento e Matematicas, Uni ersia Autonoma Metropolitana, Iztapalapa, Apartao , Mexico, D.F Mexico Receive May, 997; accepte September 9, 998. INTRODUCTION The general problem of fining solutions of linear functional equations is unoubtely one of the main problems of applie mathematics. Among such equations the classes of linear ifferential equations an linear ifference equations are very important an numerous theories an methos for their solution have been evelope. The methos base on integral transforms an the operational methos are certainly the ones use most often. In the present paper we present some general linear algebra results that may be consiere as a unifie approach to the transform an operational methos. We use the linear algebra founations of the Laplace transform an the concept of similarity to generalize a escription of the general solution of linear ifferential equations with constant coefficients in terms of certain basis for the space of exponential polynomials, an a convolution prouct efine in an algebraic way using the basis. The generalization gives an explicit construction of the general solution of any * Research partially supporte by a fellowship from CONACYT-Mexico. Research partially supporte by a grant from CONACYT-Mexico. vere@xanum.uam.mx $30.00 Copyright 999 by Acaemic Press All rights of reprouction in any form reserve.

2 30 ELIZARRARAZ AND VERDE-STAR linear functional equation that is similar to a ifferential equation with constant coefficients. Let G be a complex vector space an let L: G G be a linear operator. Let uz be a monic polynomial of positive egree an let f be a given element of G. Consier the problem of fining all the solutions g in G of the linear equation, už L. g f. Ž.. In Section 4 we show that solving Ž.. is quite easy provie that we can fin a basis for G that is relate in a certain way to the operator L. See Eq. Ž This is a way of saying that L behaves as a generalize ifferentiation operator on the space G. The construction of the solutions of Ž.. requires the introuction of a convolution prouct in the space G. Such a prouct is efine in a purely algebraic way in terms of the basic elements of G. The convolution is use to construct a right inverse for the operator ul. Our motivation comes from the linear algebra ieas that serve as a founation for the transform methos for the solution of linear equations. In Section we use a simplifie algebraic version of the Laplace transform to illustrate how we use the iea of similarity in orer to obtain the abstract setting for our main results. In Section 3 we present some elementary properties of polynomials an rational functions, relate to partial fractions ecomposition, an we use them to escribe the general solution of an inhomogeneous linear ifferential equation with constant coefficients. The results of Section 3 are use as a moel which we translate to an abstract setting in Section 4. We also give irect proofs of the main results in the abstract setting, without relying on similarity or particular properties of the vector space or the linear operators. In Section 5 we use similarity to show that our general methos can be applie to a large class of linear ifferential equations with variable coefficients. In Section 6 we apply the methos to equations of the form ul g f where L at D btiis a linear ifferential operator of first orer with variable coefficients. In Section 7 we present several concrete examples an some criteria to etermine if a given secon-orer ifferential operator with variable coefficients can be written in the form ul. Among the examples we inclue the so-calle binomial linear ifferential equation of orer n, which is relate to the Hermite polynomials. The first part of this paper presents a significant simplification of the evelopments reporte in our previous papers 6 9. It also clarifies the role of convolutions in the classical transform an operational methos, an simplifies the construction of convolution proucts that relate in a

3 LINEAR DIFFERENTIAL OPERATOR 3 nice way to a given operator, which is one of the main problems in Dimovsi s boo 3. The secon part contains generalizations of some results presente in 4.. SIMILARITY AND EQUIVALENT EQUATIONS Let X an Y be nonempty sets an let T: X Y be a biective function. Let f: X X be a function an let b be a given element of X. Define S x X: fž x. b 4. Let F: Y Y be efine by F T f T an let U y Y: FŽ y. TŽ b.4. Using only basic properties of the composition of functions it is easy to see that S T Ž U. an U TŽ S.. Therefore, fining the solution set S is equivalent to fining the set U. In other wors, solving fž z. b in the set X is logically equivalent to solving FŽ y. TŽ b. in the set Y. In some concrete situations one of the two problems is consiere easier than the other one. For example, in orer to solve fž x. b one applies the transform T to get the transforme equation FŽ y. TŽ b., which is consiere easier to be solve. Then one fins somehow the solution set U, an finally, the set S is obtaine by applying the inverse transform T to the elements of U. This is the founation of the transform methos, lie the Laplace transform for ifferential equations an the z-transform for ifference equations. Let us note that such ieas yiel a useful general metho only if the computation of images uner T an T can be one in a relatively simple way. We use next an algebraic version of the Laplace transform to illustrate the main ieas of the present paper. Define the functions, t at e t e, a,, a,! where t is a complex variable, an let E be the complex vector space generate by the set of all the e, for Ž a,. a, in. The elements of E are calle quasi-polynomials or exponential polynomials. Let R be the complex vector space spanne by the rational functions, ra, Ž z., a,. Ž z a. The elements of R are calle proper rational functions. By the ivision algorithm for polynomials the space of all the rational functions is the irect sum P R, where P enotes the space of all polynomials in one variable. The biective map T between the bases of E an R, efine by Te r, extens by linearity to a vector space isomorphism T: E R. a, a,

4 3 ELIZARRARAZ AND VERDE-STAR The usual ifferentiation operator D satisfies ae a,0, if 0, Dea, ½ Ž.. ae a, e a,, if. Let us efine the linear operator H on the space R by ar a,0, if 0, Hra, ½ Ž.. ar a, r a,, if. It is clear that D T HT. Thus for any polynomial u the ifferential operator ud, which acts on the space E, is similar to the operator uh, which acts on R. A trivial computation yiels Hr Ž z. zr Ž z. a, a,, for, an Hr Ž z. a,0 zra,0. This shows that the action of H on the proper rational functions is multiplication by z followe by proection on the space R. Thus, for g in R an 0, H gž z. is the proper rational part of the rational function z gž z.. In other wors, H gž z. is z gž z. reuce moulo the polynomials. Let u be a monic polynomial of positive egree an let f be a given element of E. Then, solving the ifferential equation, už D. y f, Ž.3. in the space E is equivalent to fining the proper rational functions g that satisfy the equation uh gz Tf, which may be written in the form, It is clear that the set of solutions of.4 is už z. gž z. Tf Ž mo P.. Ž.4. ½ 5 Tf p p U g : p P an R. Ž.5. u u u Notice that because Tf is in R then Tf u is also in R. Note also that p u is in R if an only if p is a polynomial whose egree is strictly smaller than the egree of u. The usual Laplace transform metho gives the set of solutions of Ž.3. in the form S T Ž U.. The main tool for the computation of images uner T is the partial fractions ecomposition formula. Once we have an element g of U written as a linear combination of basic proper rational functions, applying T is ust substitution of each ra, that appears in the expression for g by the corresponing basic exponential polynomial e. a, Let us note that the proceure escribe previously to fin the solutions of.3 oes not require the representation of the map T as an integral

5 LINEAR DIFFERENTIAL OPERATOR 33 transform. Note also that one of the main limitations to the generality of Eq. Ž.3. is the conition that the forcing function f must be an element of E. This limitation will be eliminate in the next section. We will fin first a escription of the construction of the set U in terms of operations with basic proper rational functions, an then use similarity to translate the construction to one that can be one entirely in the space E, without using the maps T an T. The logical equivalence of Eqs. Ž.3. an Ž.4. maes this obective possible. 3. PARTIAL FRACTIONS AND THE CONVOLUTION ON E In this section we introuce some efinitions an basic properties of rational functions that will be use in the rest of the paper. In orer to express a proper rational function as a linear combination of basic functions r a,, we nee the partial fractions ecomposition formula, which we iscuss next. Let Ł m už z. Ž z a., Ž where the a are istinct complex numbers, the m are positive integers, an the egree of u is Ým n. Define the polynomials už z. q, m Ž z., 0, 0 m. Ž 3.. Ž z a. Let T a, enote the Taylor functional efine by D fž a. Ta, fž z., Ž a,., Ž 3.3.! where f is any function for which the right-han sie is well efine. The linear functionals Q on the space P are efine by, pž z. Q, pž z. T a,, 0, 0 m. Ž 3.4. q z,0 A simple computation using Leibniz s rule yiels the biorthogonality relation, Q, qi, m Ž,., Ži, m.. Ž 3.5.

6 34 ELIZARRARAZ AND VERDE-STAR Therefore the polynomials qi, m form a basis for the vector space Pn of the polynomials with egree at most equal to n, an the functionals Q, form a basis for the ual space of P n. Consequently, for every polynomial p in P we have n m Ý Ý,, pž z. Q pq Ž z.. Ž Diviing by uz an reorering the terms in the inner sum we get the partial fractions ecomposition formula, m pž z. Q, m p Ý Ý. Ž 3.7. už z. Ž z a. 0 0 Because the elements of the set U, efine in Ž.5., are of the form g Ž Tf. u p u, where p is in P n, the partial fractions ecomposition formula Ž PFD. gives us p u as a linear combination of basic proper rational functions. The term Ž Tf. u is a prouct of elements of R. The PFD can be use to express it as a linear combination of basic rational functions. Taing uz Ž z a. Ž z b. m, with a b, an applying the PFD we get the multiplication formula, Ý r r CŽ a, ; b, m. r CŽ b, ; a,. r, Ž 3.8. a, b, m a, b, m 0 0 where the coefficients are efine by ž / i i CŽ a, ; b, i. Ž. Ž a b., a b, i,. Ž 3.9. Notice that ra, ra, m r a, m. Taing p in the PFD formula Ž 3.7. we get m m Ý r z, 3.0 už z. 0 0 Ý Ý, a, where Q. Ž 3..,, m Note that the coefficients multiplicities., epen only on the roots of uz an their

7 LINEAR DIFFERENTIAL OPERATOR 35 Multiplying 3.0 by uz an using 3. we get the polynomial ientity, m Ý Ý, q, m Ž z.. Ž Now we can escribe the construction of the elements of the set U in terms of basic proper rational functions. Because f is in E it is a finite linear combination of functions of the form e Ž. b, t an hence Tf is a linear combination of functions r Ž z.. Therefore every solution g of Eq. Ž.4. b, has the form, m Ý Ý m Ý Ý gž z. r Ž z. Tf Ž z. r Ž z., Ž 3.3., a,, a, where the coefficients, are arbitrary complex numbers. We want to transfer 3.3 to the corresponing equation for T g in the space E. Instea of performing first the multiplication in the first term of 3.3 an then apply T, we can efine an operation on E that correspons to multiplication in R. The convolution on E is efine by Ý e e CŽ a, ; b, m. e CŽ b, ; a,. e, Ž 3.4. a, b, m a, b, m 0 0 where the coefficient functions are efine in 3.9. This efinition clearly implies m Ý f h T Ž TfTh., f, h E, Ž 3.5. where the multiplication in the right-han sie is multiplication of rational functions. Now we can apply T to Ž 3.3. to obtain the solutions of Eq. Ž.3.. In this way we get the following. THEOREM 3.. The general solution of the ifferential equation už D. yž t. fž. t, where f is a gi en element of E is m Ý Ý yž t. fž t. h Ž t. e Ž t., Ž 3.6. u, a, 0 0 where m Ý Ý h Ž t. e Ž t., u, a, 0 0 the are efine in 3., an the are arbitrary complex numbers.,,

8 36 ELIZARRARAZ AND VERDE-STAR In orer to compute the right-han sie of Ž 3.6. we only nee the representation of f as a linear combination of basic exponential polynomials, the etermination of the numbers,, which epen only on the polynomial u, an the convolution formula Ž All of this can now be one without ever mentioning the map T. Notice that the last term in Ž 3.6. is the general solution of the homogeneous equation udy 0, an the other term is a particular solution of the inhomogeneous equation udy f. Note also that h u T Ž u.. COROLLARY 3.. The linear map on the space E that sens f to f h is a u right in erse for the operator u D. 4. THE ABSTRACT SETTING In this section we use the results about the space E of exponential polynomials presente in the previous sections as a moel an we obtain extensions of the main results in a very general setting. Although the use of similarity gives an almost trivial proof of our results, we prefer to give irect proofs that use only basic properties of polynomials an rational functions, an which are also useful to obtain important generalizations. Let G be a complex vector space that has a basis g : Ž a,. 4 a,. We efine the commutative convolution prouct on G as follows. If a b, Ý g g CŽ a, ; b, m. g CŽ b, ; a,. g, Ž 4.. a, b, m a, b, m 0 0 where the coefficient functions are efine in 3.9, an m Ý ga, gb, m g a, m,, m. Ž 4.. The linear map L: G G is efine by ag a,0, if 0, Lga, ½ Ž 4.3. ag a, g a,, if. For any a in it is obvious that an thus, 0, if 0, Ž L ai. ga, ½ g, if, Ž 4.4. a, m Ž L ai. ga, 0, m. Ž 4.5.

9 LINEAR DIFFERENTIAL OPERATOR 37 By inuction on m it is easy to see that s ž / m m m Ž L ai. gb, Ý Ž b a. g, Ž 4.6 b,. 0 4 where s min m,. Therefore, for a b we have m Ž L ai. g 0, m,. Ž 4.7. b, A straightforwar computation yiels LŽ ga, gb, m. Ž Lga,. gb, m gb, m g a,, Ž 4.8. where is the linear functional on G efine by g. By a, 0, linearity we obtain LŽ g f. Ž Lg. f f g, f, g G. Ž 4.9. Using inuction it is easy to prove that Ž L ai. Ž ga, f. f, Ž a,., f G. Ž 4.0. Let uz be a monic polynomial of positive egree as in Ž 3... Then, by Ž 3.0., the PFD for u is We efine m Ý Ý, r a, Ž z.. už z. 0 0 m Ý Ý h g. Ž 4.. u, a, 0 0 Translating by similarity Theorem 3. to the present setting we obtain THEOREM 4.. Let f be a gi en element of G an let the polynomial u be as in the pre ious text. Then the general solution of the equation už L. g fis m Ý Ý g h f g, Ž 4.. u, a, 0 0 where h is efine in Ž 4.. u an the coefficients, are arbitrary complex numbers.

10 38 ELIZARRARAZ AND VERDE-STAR Proof. The efinition 3. of the polynomials q gives, už z. q Ž z.ž z a., 0, 0 m,, m an thus we have the factorizations, m u L q L L ai, 0, 0 m. Then, by 4.5 we have Ž 4.3. už L. g 0, 0, 0 m, Ž 4.4. a, an by 4.0, Ž. Ž a,. L ai g f f. Let us note that Ž 4.7. an Ž 4.4. imply that g a, :0, 0 m 4 is a basis for the ernel of ul. Therefore for any g given by Ž 4.. we have ulg ul Ž h f. an hence, u m Ž u. Ý Ý, Ž a,. u L h f u L g f 0 0 m Ý Ý,, m a, q L L ai g f 0 0 m Ý Ý,, m q L f 0 If. 0 We have use here the polynomial ientity 3.. Suppose now that the convolution can be extene to a commutative bilinear map from G K to K, where K is a complex vector space that contains G. Let us note that Ž 4.9. allows us to exten the efinition of the operator L to the set of elements of the form g f, where g is in G an f is in K. From the proof of the theorem it is clear that the computation that yiels ul Ž h f. u f oes not require the application of L to f. If g is an element of G such that g 0 an f is in K then L m Ž g f. Ž L m g. f an thus, wž L.Ž g f. Ž wž L. g. f, w P.

11 LINEAR DIFFERENTIAL OPERATOR 39 By a property of the resiues of rational functions, for any polynomial u with egree greater than or equal to we have Ž u. 0 an hence h 0. Because Lg Ž f. ag f f it is clear that L Ž g f. u a,0 a,0 a,0 is efine if an only if Lf is efine. These observations about the extensions of show that we can we can generalize Theorem 4. an we can obtain the following. COROLLARY 4.. Let f be a gi en element of K an let u be as in the preceing text. Then the general solution of the equation už L. g fis m Ý Ý g h f g, Ž 4.5. u, a, 0 0 where hu is as earlier an the, are arbitrary numbers. Note that the element hu is a sort of Green s function for the operator ul. The familiar example of the space of exponential polynomials will be use to illustrate how the convolution can be extene to a larger space. Let G enote the space of exponential polynomials of a real variable t. For f an g in G the Duhamel con olution is efine by H t f gž t. fž x. gž t x. x. Ž In 6, Theorem 4. we prove that the convolution on the space of exponential polynomials efine in Ž 3.4. coincies with the Duhamel convolution. Let us enote by K the vector space of all the piecewise continuous functions of the real variable t. It is clear that G is containe in K an that the convolution efine in Ž 4.6. is well efine for f an g in K. Therefore Corollary 4. hols an Ž 4.5. gives the general solution of equations of the form udg f, where f is an element of K. Notice that in this case f fž. 0, for f in K. Note that the integral representation Ž 4.6. is what allows us to exten the convolution to a larger space of functions. The approach that we use in this section can be summarize as follows. We begin with a vector space G that has a basis g 4 a, an then we efine in Ž 4.3. the operator L in terms of its action on the basis. Another approach is to start with a linear operator L that acts on certain space of functions of a variable t an then to construct the space G an the basis g 4 for which Ž 4.3. hols. One way to o this is to fin a function GŽ z, t. a, of two variables, efine on some suitable omain, that satisfies LGŽ z, t. zgž z, t., Ž 4.7.

12 40 ELIZARRARAZ AND VERDE-STAR where L acts with respect to t, consiering z as a parameter. Then it is easy to see that the functions, ga, Ž t. Ta, GŽ z, t., Ž a,., Ž 4.8. where the Taylor functional acts with respect to z, satisfy conition Ž 4.3. an generate a vector space G. The function GŽ z, t. is calle a generating function for the operator L. For example, for the ifferential operator D a generating function is expž zt., which yiels the basic functions e a,. Using generating functions we can stuy other ins of operators, lie ifference operators, for example. See our previous papers 6, 7, an SIMILAR OPERATORS AND CONVOLUTIONS Using the space of exponential polynomials an the concept of similarity we can construct many other interesting examples. Suppose A: H G is a linear isomorphism from a space H, of functions efine on some subset of the real line, onto the space G of real exponential polynomials. Then the operator L A DA maps H into H an the functions ha, A ea, form a basis for H. Denote by the convolution on H. Then we have h f A Ž Ah Af., h, f H, Ž 5.. where enotes the convolution on G. Note that for each appropriate operator A Eq. Ž 5.. gives also an integral representation for. It is clear from the efinitions that LŽ h f. Ž Lh. f f h, h, f H, Ž 5.. where the functional on H is efine by h Ž Ah. Ž Ah.Ž 0,. because the functional on G is evaluation at zero. We will use the same symbol to enote the corresponing functional on any space isomorphic to E. We consier next an important class of examples relate to the iea of change of variables in ifferential equations. Let Ž. t be a function efine on an interval J Ž c,. that contains zero, such that Ž t. has an inverse uner composition that we enote by Ž t.. Let Ž. t be a function efine on J such that Ž. t 0 for t in J. Denote by S the operator of substitution of t by Ž. t, an by M the operator of multiplication by Ž. t. Then the operator L efine by L M S DS M Ž 5.3.

13 LINEAR DIFFERENTIAL OPERATOR 4 is similar to the ifferential operator D. In this case the basis for the space H is the set of functions, Ž t. h M S e exp a a, a, Ž Ž t... Ž 5.4. Ž t.! From 5. we see that the convolution associate with the operator L is 4 f h M S S M f S M h, where is the Duhamel convolution. Therefore, t H Ž. Ž. Ž f h.ž t. Ž t. Ž y. f Ž t. Ž y. Ž t. Ž. 0 Ž y. hž y. Ž y. y. Ž 5.5. This integral representation can be use to efine the convolution of functions in a space much larger than H. Using some basic properties of the Duhamel convolution it is easy to prove the following. PROPOSITION 5.. Let an be as in the foregoing text, let f be a piecewise continuous function on the inter al J Ž c,., an let h be a continuously ifferentiable function on J. Then we ha e Ž. where 0. 0 LŽ f h.ž t. fž t. LhŽ t. fž t. hž 0. Ž 0., Ž 5.6. Notice that we can apply Corollary 4. to fin the solutions of equations of the form ulg f, where L is an operator of the form Ž 5.3. an f is as in Proposition GENERAL LINEAR DIFFERENTIAL OPERATOR OF THE FIRST ORDER In this section we consier some elementary properties of the general linear first-orer ifferential operator L, efine by L až t. D bž t. I, Ž 6.. where at Ž. an bt Ž. are continuous functions on an interval J Ž c,., an at Ž. 0 for t in J. A simple calculation yiels the following.

14 4 ELIZARRARAZ AND VERDE-STAR Ž. Ž. PROPOSITION 6.. Let r t an s t be ifferentiable functions efine on J such that bž t. r Ž t., s Ž t.. Ž 6.. až t. až t. Ž. Then the function G z, t exp zs t r t satisfies From the generating function G z, t LG Ž z, t. zgž z, t.. Ž 6.3. we obtain the functions, Ž sž t.. r Žt. asžt. ga, Ž t. Ta, GŽ z, t. e e, Ž a,.. Ž 6.4.! These functions generate a vector space G an satisfy ag a,0, if 0, L ga, ½ Ž 6.5. ag a, g a,, if. Therefore we can apply Theorem 4. to solve equations of the form ul g f, for f in G. We show next that L is similar to the ifferential operator D. If L is an operator similar to D given by Ž 5.3. then a simple computation gives Ž t. LyŽ t. Ž Ž t.. y Ž t. Ž Ž t.. yž t., Ž t. that is Ž t. L D I. Ž t. Ž t. Ž t. From this expression an 6., we obtain immeiately the following. PROPOSITION 6.. Let L be the ifferential operator efine by Ž 6.. an let rž. t an sž. t be functions that satisfy Ž 6... Suppose that sž. t has an in erse uner composition enote by sž. t. Then L can be written as Ž. Ž. Ž. ržt. where t s t an t e. L M S DS M, Therefore the operator L is similar to D.

15 LINEAR DIFFERENTIAL OPERATOR 43 COROLLARY 6.. If sž t. 0 0 for some t0 in J, then the con olution for the operator L is an H f h t r Žt. e t exp r s s t s y t 0 Ž Ž.. 4 Ž. f s sž t. sž y. e rž y. hž y. s Ž y. y, Ž 6.6. LŽ f h.ž t. fž t. LhŽ t. fž t. hž t0. e ržt 0., Ž 6.7. for h in G an some appropriate function f. 7. SOME EXAMPLES Let uz z b z b be a monic polynomial of egree an let L be the ifferential operator efine by Ž 3... Then 4 4 už L. a Ž t. D až t. a Ž t. bž t. b D b t a t b t bb t b I. 7. Using this equation we characterize two types of ifferential equations that can be solve by the metho escribe in the previous sections. First, if we put at Ž. in Ž 7.. an we compare it with the linear ifferential equation of secon orer, we obtain the following. y Ž t. pž t. y Ž t. qž t. yž t. 0, Ž 7.. PROPOSITION 7.. The ifferential Eq. Ž 7.. can be written in the form ul 0, where u z z b z b, L D bt, Ž. an bž. t Žpt Ž. b., if p Ž t. p Ž t. 4qŽ t. b 4b. Ž. In an analogous way, taing bt 0 in 7. we get Ž. PROPOSITION 7.. Let L a t D an let u z z bz b be a monic polynomial of egree with b 0. A necessary an sufficient conition for the equation, y Ž t. fž t. y Ž t. g Ž t. yž t. 0,

16 44 ELIZARRARAZ AND VERDE-STAR to be of the form u L 0 is Ž. Ž. In such case a t b g t. EXAMPLE. fž t. gž t. g Ž t. b. g Ž t. b For, in, let us consier the ifferential equation, y Ž t. y tž t. y 0, t. Ž 7.3. Ž. Ž. In this case pt t, qt t t, an p Ž t. p Ž t. 4qŽ t. 4. Therefore it is possible to apply Proposition 7. to solve the equation. In orer that b 4b 4, we can select the values of b an b in Ž. Ž several ways. Let b 0, b 4, uz z 4., an L D Ž t.. Then Eq. Ž 7.3. taes the form uly 0. Therefore its general solution is given by ½ ž / 5 ½ ž / 5 yž t. C exp at 0 t C exp at t, where a, a are the roots of the polynomial už z,. 0 an C an C are arbitrary complex numbers. EXAMPLE. For the ifferential equation, ty Ž t. y t 3 y 0, 0 t, Ž 7.4. Ž. Ž. if we let f t t t, an gt t, then fž t. gž t. g Ž t.. g Ž t. This means that the hypothesis of Proposition 7. is satisfie. We choose b an b such that b b, for example, b b. Then the ifferential equation can be written as uly 0, with L Ž t. D an uz z z 4. Therefore the general solution of Ž 7.4. is ž / ž / t t yž t. C exp a0 C exp a,

17 ' LINEAR DIFFERENTIAL OPERATOR 45 ' where a0 3 i, a 3 i, an C, C are any complex numbers. Consier now the inhomogeneous equation, Ž t. y y t y t, t 0. Ž 7.5. t For the operator L t D the corresponing convolution is t H ž ' / 0 Ž h F.Ž t. F t y hž y. yy. Ž 7.6. Because uz z z 4, a simple computation gives ' u ½ ž / ž / 5 a a 0 h Ž t. exp t exp t, where 3 6 i. Let Ft t, the right-han sie of 7.5. Then a particular solution for Ž 7.5. is u ½ ž / ž / 5 t a a 0 h F t H t y exp t exp t yy. 0 Using integration by parts it is not ifficult to obtain ½ ž / ž / 5 a a 0 Ž hu F.Ž t. Ž t. a0 exp t a exp t. 4 8 In a similar way we can solve the following ifferential equations, y ty e t y 0, 3 y tan t y sec t y 0, 4 ž / y y y 0, t 4t y Ž cos t. y sin tž sin t. y 0. 4 For more examples of this in see 4.

18 46 ELIZARRARAZ AND VERDE-STAR EXAMPLE 3. We consier now a class of linear ifferential equations that are easily recognize. For Ž c, n. in, the equation, n n n Ý ž / Ž ct. D y 0 Ž is calle the binomial linear ifferential equation of orer n. This equation has been stuie by several authors. See, 5, an 0. Let He Ž z; c. enote the Chebyshev Hermite polynomials, efine by n n n n! c HenŽ z; c. Ý z. Ž n.!! 0 Then we can prove the ientity, n n n n Ý ž / 0 He D cti; c ct D, 7.8 which generalizes some operator ientities of Klamin 5 an Chatterea. This means that the ifferential operator of Ž 7.7. is a polynomial in the operator L D cti. By Proposition 6. the generating function for L is c ž / G z, t, c exp zt t, an hence the functions g Ž. t GŽ, t, c., for,,..., n, constitute a funamental system of solutions for the ifferential equation Ž 7.7., where,,..., n, are the zeros of the Chebyshev Hermite polynomial He Ž z; c. n, which are istinct. Let us note that any ientity of the form, n už a t D b t I. Ý f n t D, where u is a polynomial an a, b, an f are functions of t, allows us to solve ifferential equations of the form wž L. y 0, where L is the operator in the right-han sie of Eq. Ž 7.9. an w is a polynomial. Several ientities of this in have been obtaine in an.

19 LINEAR DIFFERENTIAL OPERATOR 47 REFERENCES. S. K. Chatterea, On the n-th orer linear ifferential equation an Hermite polynomials, Yoohama Math. J. 6 Ž 968., S. K. Chatterea an H. M. Srivastava, Operational representations for the classical Hermite polynomials, Stu. Appl. Math. 83 Ž 990., I. H. Dimovsi, Convolutional Calculus, n e., Kluwer Acaemic, Dorrecht, D. Elizarraraz an L. Vere-Star, On a class of ifferential equations that contains the equations of Euler an Chebyshev, A. in Appl. Math. 9 Ž 997., M. S. Klamin, On the n-th orer linear ifferential equation, Amer. Math. Monthly 74 Ž 967., L. Vere-Star, Solution of linear ifferential equations by the metho of ivie ifferences, A. in Appl. Math. 6 Ž 995., L. Vere-Star, An algebraic approach to convolutions an transform methos, A. in Appl. Math. 9 Ž 997., L. Vere-Star, A Hopf algebra structure on rational functions, A. in Math. 6 Ž 995., L. Vere-Star, Divie ifferences an linearly recurrent sequences, Stu. Appl. Math. 95 Ž 995., O. V. Visov, Linear ifferential equations of binomial form, Math. Notes 5 Ž 99., O. V. Visov an H. M. Srivastava, New approaches to certain ientities involving ifferential operators, J. Math. Anal. Appl. 86 Ž 994., 0.