On a transformation of solutions of the differential equation \(y''=Q(t)y\) with a complex coefficient Q of a real variable. (English) Zbl 0712.34009

The author considers differential equations of the following type: (*) \(y''=Q(t)y\) where Q: \({\mathbb{R}}\to {\mathbb{C}}\) with Im Q(t)\(\not\equiv 0\) and Q continuous. A function X: \({\mathbb{R}}\to {\mathbb{R}}\) is called a (complete) transformator of (*) if (i) \(X\in C^ 3({\mathbb{R}})\), \(X'(t)\neq 0\) for \(t\in {\mathbb{R}}\), \(X(R)=R\), (ii) for every solution y(t) of (*) the function given by y(X(t))/\(\sqrt{| X'(t)|}\) is a solution of (*) again. It is shown, that X is a transformator of (*) exactly if \(X({\mathbb{R}})={\mathbb{R}}\) and X is a solution of the following system of differential equations: \[ - \frac{X'''(t)}{2X'(t)}+\frac{3}{4}(\frac{X''(t)}{X'(t)})^ 2+(X'(t))^ 2\cdot Re Q(X)=Re Q(t);\quad (X'(t))^ 2\cdot Im Q(X)=Im Q(t). \] Some special properties of X are shown and the so-called central transformator is introduced. The set of transformators of (*) (resp. the set of increasing transformators of (*)) constitutes a group relative to the rule of composition of functions; this group \(L_ Q\) (resp. \(L_ Q^+)\) is extensively studied.
Reviewer: H.Ade


34A30 Linear ordinary differential equations and systems
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