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Summary:
The article studies a general linear differential equations of the 2nd order the solution $y(t)$ of which generates an orthogonal sequence $y(k_it)$ where $k_i$ is a suitably ordered sequence of positive roots of the solution $y(t)$. Actually differential equations of Euler's type are dealt with.
Further a certain generalization concerning the general solution of certain iterated equation of the 4th order is given.
References:
[1] B. G. Korenjev: Some elasticity and heat conduction problems solvable in Bessel functions. (Russian), Moscow 1960.
[2] V. Pane: Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes. Aplikace matematiky, sv. 16 (1971), čís. 3.
[3] A. Gray G. B. Mathews: A treatise on Bessel functions and their applications to physics.
[4] H. Bateman: Higher transcendental functions. Vol. 2. Zbl 0542.33002
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