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 ...

116 Krystyna Grytczuk 2. Proof of the Theorem. Putting in (2) u

k(x) = x , So,*:^) = 1 for k = 1,2, • • •, n we get (6) Vk = Vo = x

x. By the définition of the functions s

m>

k(x) and (6) we obtain (7

) I Sj(x)A(A - 1)... (A - (i - 1)) j = 1,2, - - -, n By (4) and (7) is follows that (8) Po(z)s

n(x) + Pi(®)a

n_i(®) + • • • + P

n(x)s

0{x) = 0 . On the other hand from (1) we have (9) Po(x) = x

n , A(x) = a!^"

1

, • • •, Pn-1 (z) = x

n , P

n{x) = a

n . From (7), (8) and (9) we obtain x

nA(A - 1) • • - (A - (n - 1)) x

x~

n + aix

n~

l\(\ - 1) • • • • • • (A - (n - 2)) x

A(

nx

) + • • • + a

nz

A = 0 . Let F(X) — A(A - 1) • • • (A - (n - 1)) + + aiA(A - 1) • • • (A - (A - 2)) + • • -a

n_iA + a

n . Then by (10) and (11) we have (12) F(X)x

x = 0. Since x / 0 on J then by (12) we get F(A) = 0, so denote that A must be a root of the algebraic équation A(A - 1) • • • (A - (n - 1)) + aiA(A - 1) • • • (A - (A - 2)) + h a

n-1A -f- a

n = 0 and the proof is complété.