LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE …

1 Dipartimento EnergiaLINEAR DIFFERENTIAL EQUATIONSWITH VARIABLE COEFFICIENTSF undamental Theorem of the Solving KernelCORRADO GIANNANTONIC entro Ricerche della Casaccia, RomaRT/ERG/95/07J .ENTE PER LE NUOVE TECNOLOGIE,L'ENERGIA E L'AMBIENTED ipartimento EnergiaLINEAR DIFFERENTIAL EQUATIONSWITH VARIABLE COEFFICIENTSF undamental Theorem of the Solving KernelCORRADO GIANNANTONIC entro Ricerche della Casaccia, RomaRT/ERG/95/07 Testo pervenuto nel giugno 1995I contenuti tecnico-scientifici dei rapporti tecnici dell'ENEA rispecchiano l'opinione degli autori e non necessariamente quella dell' present work shows a new fundamental theorem concerning the explicit solution of ahomogeneous or non- homogeneous LINEAR DIFFERENTIAL equation of order n, with variablecoefficients.

2 In fact the explicit solution of the mentioned EQUATIONS is reduced to theknowledge of just one particular integral: the "kernel" of the homogeneous or of theassociated homogeneous equation of its several and relevant consequences, the theorem has been properly named:Fundamental Theorem of the Solving Kernel.[ LINEAR DIFFERENTIAL EQUATIONS , VARIABLE Coefficients, Solving Kernel, Explicit Solution]SommarioII lavoro presenta un nuovo fondamentale teorema riguardante la soluzione esplicita di unaequazione differenziale lineare omogenea o non omogenea di ordine n, a coefficientivariabili, infatti, la soluzione esplicita delle citate equazioni viene ricondotta alla conoscenzadi un solo integrale particolare: il "nucleo" della equazione omogenea o, rispettivamente,della omogenea causa delle sue molteplici e rilevanti conseguenze, il teorema stato propriamentedenominato: teorema fondamentale del nucleo PAGE(S) left BLANK.

3 1 Contents1. Introduction2. Fundamental theorem of the solving kernel3. Examples4. General considerations5. Conclusions6. References16p. 1819p. 20 NEXT PAGE(S) left BLANK. 1 LINEAR DIFFERENTIAL EQUATIONS with VARIABLE CoefficientsFundamental Theorem of the Solving Kernel1 IntroductionIt is well known that the general solution of a homogeneous LINEAR DIFFERENTIAL equation of order n,with VARIABLE coefficients, is given by a LINEAR combination of n particular integrals forming afundamental set, their Wronskian is different from zero. Moreover the equation kernel can bedefined through the mentioned set of particular integrals and their present work shows a new theorem which is somehow the reversal of the previous one.

4 It statesthat if we know the kernel of a homogeneous LINEAR DIFFERENTIAL equation of order n, with variablecoefficients, we can easily find a fundamental set of particular integrals, the general solution ofthe considered is also known that the general solution of a non- homogeneous LINEAR DIFFERENTIAL equation of ordern, with VARIABLE coefficients, is given by the general solution of the corresponding homogeneous oneplus a particular integral of the non- homogeneous equation. This particular integral can be definedthrough the kernel of the corresponding homogeneous equation and the source term. Thus also thegeneral solution of the considered non- homogeneous equation can be given through the kernel of theassociated homogeneous are the reasons for having properly called the kernel as "solving" kernel and named thetheorem, because of its several and relevant consequences, as "Fundamental" Theorem of theSolving Fundamental theorem of the solving kernelSuppose we have a homogeneous LINEAR DIFFERENTIAL equation of order n, with VARIABLE coefficients^^f^ O (1)and its associated initial conditions given by/(A)(0) = /A , k = 0,1,2, ,n-l.

5 (2)Let K{t, T) be the solving kernel of the homogeneous equation ( that particular integral ofequation (1) which satisfy the initial conditions/w(r) = 0 for k = 0,1,2, ,n-2 and/'"~'V) = ' ) We are going to demonstrate the following theorem:Theorem. The general solution of equation (I) can be expressed through the solving kernel K(t, r)and its derivatives up to the order n-1, and through the equation coefficients and their derivatives,under the only hypothesis thatao(/)eC , fl,(r)eCw (k = 1,2, ,n-l). (3)Proof! It can be divided into three parts:a) equation ( 1 ) re-written in an equivalent form;b) solution of the re-written equivalent equation;c) explicit solution in terms of K(t,t) and its derivatives, equation coefficients and their ) Equation (1) re-written in an equivalent is possible to re-write equation (1) in an equivalent form by introducing a new LINEAR operator D;ldefined as )f(T)dT , V/>0 (4) as convolution of the function /(/) and the j-th derivative of the Dirac delta function S(t).}

6 ' " dtJIn fact, for any j > 0, the derivative -~j- can be written asand, through equation (5), equation (1) can be re-written as follows+-+a,(/)D0 +ao(t)]f(t) = |]/,^(/) (6)where(7)b) Solution of equation ( 1) in the equivalent form (6).The solution of equation (1) re-written in the equivalent form (6) is given byEd0 (8)wheredJfEquation (5) is obtained by anti-trasforming the Laplace Transform of j and by making use of equation (4):Ij-i8U\t-T)f(T)dr - 2>(-M~*)(O/'*)(0) (see also Ref.[4], p. 249)*=o9, r)Xk(k=0,1,2, ,n-1 ).(9)For a complete demonstration of the previous assertion it is sufficient to show that each Yk(t) issolution of equation (6) when we assume the following initial conditions:/;='/ = !

7 Fori = 0,1,2, ,n-l; i * kfor i = (t,D0)Yk(t) == % +an_]V*e[0,n-l]l(0A +at(t).(10)(II)(12)If so, the correspondent Wronskian W(t) of the particular integrals Yk{t) is always different fromzero since1000 1 0 001= 1*0(13)and consequently the set of particular integrals Yk(t) is a fundamental the mentioned purpose, we firstly demonstrate thatJ!K(t,T)]Xk(T)dT, V/ e[0,/i] ,(14)where, analagously to equation (4), the operator D\ applied to a general function /(/) is defined by\ V/>0(15)10and which reduces to >//(/) in case of r = (14) can be demonstrated by firstly noting that it is easy to generalize equation (5) to ageneral initial time / = , to get(16)which, in case of the solving kernel K(t^ r) , reduces to (for = r )L") forj = 0,1,2, ,n-l(17)D'.}

8 K(t,r)-S{t-r) forj = nbecause, by definition of "kernel", we haveKa\v,v) = 0 fork = 0,1,2, ,n-l(18)KUI)(T,T) = \ fork = nTlien, by considering the structure of a prefixed Xk(t), we note that each addend is in the form (seeeq- (7))a,(t)S(m)(t), /e[0, -l], i6[0, -l] (19)and thus, to demonstrate equation (14), it is sufficient to demonstrate that, for V/ and V/M , we V/e[0,/i]. (20)11 Now, remembering that2fOr e[a,fl] (21)for t\a,0\the first and the second member of equation (20) can be re-written as follows respectively(22)11 l J r=0and0 V J r=0 Now let us distinguish two cases: j * n and j = n:i) for j ?}

9 T /;, the second member of equation (23) can be re-written, through equation ( 17), as follows<?" d'Operator - and - can be commutated in equation (24), since the required conditions for thefunction K(t,r)a,(T) are satisfied". This corresponds to commute and Dj, in accordance toequation (17), to get, successively2 see Ref. [4], p. 2463 Because of the hypothesis on the equation coefficients at(t) and the properties of K(t, x ) , if this is thought asbeing generated by an independent fundamental set of particular integrals and their correspondent (25)The last member of equation (25) equals the second member of equation (22) and thus equation (20)is demonstrated foi j' * n.

10 Ii) for j = n, the second member of equation (23) can be re-written, through equation (17), as follows/y(26)ff" d"Operators and - can be commutated in equation (26) since K(t, r)a,(r) satisfy the requireddxm dtconditions and, at the same time,44>-">(,-r)a,(r)drn dtJthat equals(28)This is equivalent to say that - and D" commute in equation (26).So we get that the equation (26) reduces to1 see Ref. [3], Vol. Ill, 2nd part, (29)r=0 The last member of equation (29) equals the last member of equation (22) written in case of j = equation (20) is completely demonstrated and, consequently, so is equation (14).