## 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.
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