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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
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References:
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