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 ...

From (11) by easy differentation we obtain (12) = /=1,2,...,« where u, (13) s

n = s

li

i

k+A-s

l_

]

kj

L; /,£ = 1,2,...,«. uk From our assumption for fixed k = 1,2,..., n we have (14) Substituting to (14) the right hand side of (11) and (12) we get (15) P

Qs

n

k • u

x

k + P

xs

n_

x

k • u

k +• - • +P

Ai

t • u

l

k = 0. From (15) we have (16) (

Po • «V + • Vu +• • • +

pn\

k X = 0. Since by 2° we have u

k (x) * 0 then by (16) it follows that and the condition (6) is proved. For the proof of sufficiency suppose that the conditions (6) are fulfilled. Then we have (17) P

0 • ^ + P, • Vu +• • •

+

pn • V = 0 for k = 1,2,...,/2. Substituting to (17) the right hand side of the formulae (13) we obtain (18) P

0 Vi,* W, w / u, Sn-2,k ^ ' Sn-2,k ' UkJ +...+ + P

n - SoJk = 0Now, we remark that (19) {s

oyul)'=S

xyul From (19) by simple induction on / we obtain 99