Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GRYTCZUK, K., Effective integrability of the differential équation ...

Effective integrability of the differential équation Po{x)y

(n

) + Pi{x)y

{

n~

l

) + • • • + Pn(x)y = o, II. KRYSTYNA GRYTCZUK ABSTRACT. In the present paper we give an application of our resuit given in [1] to the classical Euler's differential équation. 1. Introduction Consider the classical Euler's differential équation (1) zV

n

) + aiS

B-V

n1> + • • • + an^xyV + a

ny = 0 where ü{ E R In the paper [1] it was shown (see Th.

2) that the necessary and suffcient condition for the fonctions (2) y = s

0t

k(x)ul(x)

, k = 1, 2 • • •, n to be the particular solutions of the differential équation (3) Po(x)y

{n

) + i\(s)y

(n_1

) + • • • + Pn(x)y = 0 is that n (4) ^2Pj(x)s

n_

jf

k(x) = 0 , k = 1,2, • • -,n, j=o where s

mf

k(x) = s'

m_

l k(x) + s

m-and s

mi

k(x),u

k(x) E C(

n)(J); J = (x

ux

2) C R, u

k(x) £ 0 for x E J. In the present note by using this resuit we prove the following theorem. Theorem. The necessary and sufficient condition for the function y

0 = Ï

a to be a particular solution of (1) is that the A satisfies the following algebraic équation: F(X) = A(A — 1) • • • (À — (n — 1)) + ai A(A - 1) • • • (A - (n - 2)) + • • • + a

n_

x A + a

n = 0.