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 ...

iswí +•••+ (7) n +PÁ*) =0. k=1 Let in (7) n (8) ^o = >>o (*) = 2 s

0j

c (x) ui (x) k=\ By (8) it follows that (9) y- = Ís

]

k-i4 k-1 where Uk From (9) it follows that (10) j>o° = %

si,k uk\ /=1,2,...,« where Sl,k ~ Sl-\,k + Sl-\,k Uk Substituting (8) and (10) to (7) we obtain so the function y

0 is a solution of (1) and the proof is complete. Proof of the Theorem 2. For the proof of necessity suppose that the functions (11) y

k = s

o

k - u

k

, k = 1,2,...,« are the paricular solutions of (1). 98