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 ...

For the proof of necessity we suppose that the function y

0 given in (2) is a solution of (1). Then by (2) and l°-2° we obtain / u

t \ v (7) * = Let in (7) (8) s

l

k=s'

0

k+Xs

0y

1^. uk By (7) and (8) it follows that (9) y'

0 = %\

k.u

x

k. In similar way from (9) and our assumptions 1°—3° we obtain (10) y'P = Z s,

t- u

l

k k=\ where ni (11) s

ll

c =s'

l_

}

k+X-s

l_

h

k-j

L ; /,* = l,2,...,w. u, From (10) and (11) get Po/o

n

) + W" +• • • +PJ>0 = Po Z S„X I + ( n Ar=! f n (12) +PA Zvu" \ *r=l +-+P.\Ts

o

kut 1 = 0. By (12) it follows that and the condition (3) is true. For the proof of sufficiency we suppose that the condition (3) holds. Then we have 97