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On Halphen and Laguerre-Forsyth canonical forms of linear differential equations. (English) Zbl 0729.34008

Let \(y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_ 0(x)y=0\) and \(z^{(n)}+q_{n-1}(t)z^{(n-1)}+...+q_ 0(t)z=0\) be two differential equations of the n-th order, \(n\geq 2\), with continuous coefficients, defined on open intervals \(I\subset {\mathbb{R}}\) and \(J\subset {\mathbb{R}}\) respectively. They will be referred to by the symbols \(P_ n(y,x;I)\) and Q(z,t;J) respectively. The equation \(P_ n(y,x;I)\) is said to be globally transformable into \(Q_ n(z,t;J)\) if there exist a function \(f: J\to {\mathbb{R}}\), \(f\in C^ n(J)\), \(f(t)\neq 0\) on J, and a \(C^ n\)- diffeomorphism \(h: J\to I\) such that \(z(t)=f(t)y(h(t))\), \(t\in J\), is a solution of \(Q_ n(z,t;J)\) whenever y is a solution of \(P_ n(y,x;I)\). The second order equation \(n''+p_ 0(x)n=0\) on I is said to be of type k if it is not oscillatory and k is the maximal number of zeros of each nontrivial solution on I. It is of general or special kind regarding to the existence or nonexistence of two linearly independent solutions having k-1 zeros on I. Let \(A_ n\) denote the class of equations \(P_ n(y,x;I)\) with \(p_{n-1}\) vanishing on I and \(p_{n-2}\in C^{n- 2}(I)\). It is known that every equation \(P_ n(y,x;I)\) with \(p_{n- 1}\in C^{n-1}(I)\) and \(p_{n-2}\in C^{n-2}(I)\) can be transformed into an equation belonging to \(A_ n\). The equation \(P_ n(y,x;I)\) is said to be of Laguerre-Forsyth canonical form if \(p_{n-1}\) and \(p_{n- 2}\) vanish on I. As his main result, the author proves the following statement: The equation \(P_ n(y,x;I)\in A_ n\), \(n\geq 2\) can be globally transformed into the Laguerre-Forsyth form if and only if the second order equation \(n''+p_{n-2}(x)u/\binom{n+1}{3}=0\) on I with \(p\in C^{n-2}(I)\) is of type 1. Moreover, the definition interval of the Laguerre-Forsyth form is \({\mathbb{R}}\) if and only if the above second order equation is of special kind.
Reviewer: L.Janos (Praha)

MSC:

34A30 Linear ordinary differential equations and systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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