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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