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Criterion of global equivalence of linear differential equations. (English) Zbl 0552.34009
For $$n\geq 2$$, consider ordinary linear differential homogeneous equations (1) $$y^{(n)}+ p_{n-1}(x)y^{(n-1)}+... +p_ 0(x)y=0$$ on 1 and (2) $$z^{(n)}+ q_{n-1}(t)z^{(n-1)}+... +q_ 0(t)z=0$$ on J. The equations are said to be globally equivalent if $$z(t)=f(t).y(h(t))$$, $$t\in J$$, is a solution of (2) whenever y is a solution of (1), and $$f\in C^ n(J)$$, $$f(t)\neq 0$$ on J, and h is a $$C^ n$$-diffeomorphism of J onto I. A criterion of global equivalence of the equations for $$n\geq 3$$ is derived, which is in general effective, i.e. expressible in terms of coefficients and quadratures. For the second order equations the criterion was found by O. Borůvka [Linear differential transformations of the second order (1971; Zbl 0222.34002)].

##### MSC:
 34A30 Linear ordinary differential equations and systems 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
##### Keywords:
global equivalence
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##### References:
 [1] Boruvka, Differencial’nyje Uravněnija 12 pp 1347– (1976) [2] Boruvka, Linear Differential Transformations of the Second Order (1971) [3] DOI: 10.2307/2007241 · JFM 42.0344.01 [4] DOI: 10.1515/crll.1893.111.290 [5] Everitt, Czechoslovak Math. J. 32 pp 275– (1982) [6] Kummer, Progr. Evang. Königl. Stadtgymnasiums Liegnitz 100 pp 1– (1887) [7] Hustý, Publ. Fac. Sci. Univ. J. E. Purkyně (Brno 449 pp 23– (1964) [8] Halphen, Mémoires présentés par divers savants à l’académie des sciences de l’institut de France 28 pp 1– (1884) [9] Neuman, Lecture Notes in Mathematics pp 548– (1982)
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