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 ...

Put =(-l)

/Iű

u_

1 for 7 = 1,2,...,/?. Then it suffices to prove that (29) />„=(-irzv For the proof of (29) note that from the form of the matrix A by Laplace's theorem we have (30) det A = A.o(Í • vi) + (1)

1 A.i(I, Vu

•«]?)+••• + On the other hand it is easy to see that the first row of the matrix A is a linear combination of the others and therefore (31) det A = 0. By (30) and (31) it follows that í n (32) {-l)"D

l

n = A,o From (32), (28) and the fact tnat Pj-1 =(1)

/~

1A,y-i for j =1,2,...,« we obtain (-1)" A, = and the proof is complete. 102