Smoothness as an invariant property of coefficients of linear differential equations. (English) Zbl 0702.34034

It is well-known from the work of O. Boruvka that any second order linear differential equation \(y''+p_ 1(x)y'+p_ 0(x)y=0\) with continuous coefficients on a real open interval may be transformed by means of \(z(t)=f(t)y(h(t))\) into the same type of equation with real analytic coefficients, in particular into \(z''+z=0\) on some real interval J. What about nth order equations whose coefficients have order of smoothness k? The author shows that if \(k<n-2\) the order of smoothness of the coefficients cannot be improved by such a transformation whereas in the case \(k=n-2\) a transformation may be found which makes the transformed coefficients (real) analytic.
Reviewer: M.E.Muldoon


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