Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GRYTCZUK, K., Effective integrability of the differential équation ...

Effective integrability of the differential équation 117 3. Remarks. Suppose that ail roots of the équation F(X) = 0 are distinct and real. Then the differential équation of Euler (1) has n-particular solutions of the form: Vi = Z

A

l , 2/2 = X

Á

2 , • • • ,Vn = • It is easy to see that those solutions are linear independent over J = (0,+oo). Hence in this case we obtain that the generál solution of (1) has the form: y = C

lx

X

l + C

2Z

A

2 + ••• + c

n = x

A

n. Now we can assume that ail roots of the équation F{ A) = 0 are distinct but they can be comp lex numbers. If À = a + bi is a root of F then we have XA = xa+bi = = x a gib In x = x (

CO

S(6 hl x) + i sin(6 ln x)) and we see that fonctions (13) x

a cos(ölnz) and x

asin(61na;) are real solutions of (1) and linear independent over J. Since ai £ R then exists the conjugate complex root to À, namely À = a - bi. In similar way we obtain (13). If X\ is fc-multiple root of F(A) then we have (14) F(X

1) = F'(X

1) = '-- = F^l\X

1) = 0, fW(\O/O. Then by differentiation m-time the expression F(X)x

x with respect to A we obtain From (14) and (15) it follows that the fonctions Vm = x

A

l(lnx)

m , m = 0,1, • • •, k - 1 are particular solutions of (1). Référé ne e [1] K. GRYTCZUK, Effective integrability of the differential équation Po{

x)y^ + ' • " + Pn{

x)y = 0> Acta Acad. Paed. Agriensis, Sect. Mat., 21 (1993), 95-103.