Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

GRYTCZUK, K., General solution of the differential equation ...

118 Krystyna Grytczuk Proof. Putting y(x) = \nz{x) we obtain and consequently we have (5) ,"(*)= ™ W z(x) \ z(x) Since y(x) = 1nz(z), then = z(x) and by (4) and (5) it follows that (1) can be reduced to the following form: Z"(X) .

(^(Z))

2 _ _ 2 (6

) z*(x) 2 z*(x) - X " Integrating (6) with respect to x we obtain: z"(x) n

(z'(x))

2\ 1 3 U \ ~ 2 3Í ( )

dx= 4 C

l z

l{x) z

ó[x) I o Denote by x z"(x) (z'ix))

2 v ' z

z{x) z

6{x) Then we see that the function (8) , M

=F(z{x)

) satisfies the following condition and therefore by (7) and (8) it follows that / \ z'(x) 1 o _ <

9> M =