Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

Krystyna Grytczuk: Effective integrability of the differenctial equation ...

(20) {s

0yu

x

k)

tr

) = s

l

k ui for

/ = 1

,2, n. By (20) and (17) we get (21) p

0(s

oj

e-if*)

0

0 u~

k

x + P,{s

ol

c-ut) because on J. We note that (22) y

k - s

o

k • u

k and by (21) and (22) it follows that py^+pyr'+'-'+pj^ and for k = 1,2,...,». The proof is complete. Corollaries. Corollary 1. Let K = R[P

r s

n^

k\ j = 0,1,2,...,», k-\,2,...,n denotes the ring of all polynomials of the intederminates x

j

k = P

js

n_

j

k and let be linearly independent over K. Then if the function (23) y

0 = Í^

k(x)ui(x) is a solution of (1) then the all functions (24) y

k =s

o

k-u

k for * = l,2,...,w are also the particular solutions of (1). ProoL From the assumption of the Corollary 1 and the Theorem 1 it follows that By (25) and the assumption that wfare linearly independent over K we get (25) 100