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 ...

Theorem 1. Let y

Q = }>

0(x), s

l

k(x), u

k(x) and the coefficients P.(x) of (1), where jj = 0,1,...,«, £ = 1,2,...,« satisfy the conditions: 1° \

kM, J - (x,,x

2) ciR, AGR

+. 2° u

k(x) ^

0, j>

0(*)*°> P

0(x)*0 for xeJ. 3° Pj(x) eC(J). The necessary and sufficient condition for the function (2) to be a particular solution of (1) is k = 1,2,...,«. Theorem 2. Let the assumptions 1°—3° of the Theorem 1 be satisfied. The necessary and sufficient codition for the functions ® y

k = y

k(x) = s

0

k(x).u$(x), k = 1,2,...,« to be the particular solutions of (1) is where (4) s

l

k(x) = si

x

k(x)+s^

k(x)^\ (6) ZP

J(x).s

n_

j

k(x) = 0, k - 1,2,...,« /=0 ' where (x) are as in (4). Proof of the Theorem 1. 96