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Summary:
The paper gives a comprehensive review of general solutions of the ordinary linear differential equation (1) $\Delta^2w+2\epsilon\Delta w+w=0, \ \Delta=d^2/d\rho^2+(1/\rho)(d/d\rho)$, the particular solution of which is represented by the Bessel function $w=Z_0(\rho\sqrt{\lambda})$of zero index with a real, imaginary or complex argument, respectively.
In the case $\epsilon = \pm1$ the corresponding characteristic equation $\lambda^2 - 2\epsilon\lambda + 1=0$ evidently zields one double root $\lambda=\pm 1$; then another independent particular solution of Eq. (1) is represented by the function $w=\rho Z_1(\rho\sqrt{\pm 1})$.
Generally it is proved that the general solution of a double Bessel equation of the $v$-th index (2) $[\Delta + \lambda - (v/\rho)^2]^2w=0$ can be written in the form $w=A_1J_v(\rho sqrt{\lambda})+A_2\rho J_{v+1}(\rho\sqrt{\lambda})+A_3Y_v(\rho\sqrt{\lambda})+A_4\rho Y_{v+1}(\rho\sqrt{\lambda})$ where $A_1$ to $A_4$ denote the constants of integration and $J_v(\rho \sqrt{\lambda}),\ Y_v(\rho \sqrt{\lambda})$ are the Bessel functions of the $v$-th index of the first and second kinds, respectively.
References:
[1] B. G. Korenjew: Einige in Besselschen Funktionen lösbare Aufgaben der Elastizitatstheorie und Wärmeleitung. (russisch), Moskau 1960.
[2] V. Pane: Theorie der schubweichen Kreisplatte auf elastischer Unterlage. Acta mechanica, Vol. 1/3, 1965, 294-317. DOI 10.1007/BF01387240
[3] F. Schleicher: Kreisplatten auf elastischer Unterlage. Berlin 1926.
[4] Table of the Bessel Functions $J_0 (z)$ and $J_1 (z)$ for Complex Arguments. New York 1943, Moskau 1963. Zbl 0061.30206
[5] Table of the Bessel Functions $Y_0 (z)$ and $Y_1 (z)$ for Complex Arguments. New York 1950, Moskau 1963. Zbl 0041.24504
[6] Jahnke-Emde-Lösch: Tafeln höherer Funktionen. Stuttgart 1960.
[7] E. Kamke: Differentialgleichungen, Lösungsmethoden und Lösungen. B. 1, Gewöhnliche Differentialgleichungen, Leipzig 1951.
[8] E. T. Whittaker G. N. Watson: A Course of Modern Analysis. Cambridge 1927.
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