By H. Amann (auth.), Jiří Fábera (eds.)
Read Online or Download Equadiff IV: Proceedings, Prague, August 22–26, 1977 PDF
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Extra info for Equadiff IV: Proceedings, Prague, August 22–26, 1977
Example text
8~. Hence This yields e N ~ Q+ (= ~Q) and, since • increases, • • ~ ; because • com+ mutes with every X • ~ Q we find • • ~Q. The result: ~ R c ~Q. The rest of the proof follows by similar arguments. Theorem ~. The relation ~R o R(t) - Q(t) (t • R) and vice versa. ~ Q implies [R(~) - Q(~)]~ "2(t)= 41 Proof. @ being the fundamental dispersion of Q it satisfies the equation (QQ). If ~ R D ~Q then • also satisfies the equation (RR) and the above relation follows. The rest of the proof is obvious.
8. The e~uation Y" = -Y ~t ~ ~)~ In the following formulae ~integer), a,b,c denote constants; t c ~. Integrals: Y(t) = Cosin(a+t); 0 ~ a < ~ , c ~ 0. 42 Dispersions: c(t) = v t a n _ l ( c . ~ ) ~ r ~ ± ÷ ~ ~(-a) = v~ Increasing dispersions: ; ; 0 ~ a,b < ~ ; c(b-a) ~ 0. The last formula with c(b-a) > 0. Central dispersions: ~v(t) = t + v~ Fundamental dispersion: ~(t) Co-dispersions: = t + h(t) = 6 = ZI; d E ~ ; . ~. d'(t)< The adjoint groups of -i are also denoted b y @ SO that ~ - I = @ ' ~-i = ~ ' ~ 1 = ' -i ,• , ~+, " is called the fundamental group; its elements special dispersions.
References. I. INTRODUCTI ON The theory of global properties of the ordinary 2rid order linear differential equations in the real domain has, in the last twenty years, made a remarkable progress. It originated from the problem of global equivalence of the nth order linear equations (n > 2) as the first step towards its solution. The question was, first, to solve the problem of global equivalence in the simplest case n = 2 and then, with acquired experience, get to the core of the problem in the general case.