Geodesic
Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes
Applications of Mathematics, Tome 16 (1971) no. 3, pp. 203-214

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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.
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.
DOI : 10.21136/AM.1971.103346
Classification : 30D05, 34M99, 39B22, 39B32, 74Bxx 
Panc, Vladimír. Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes. Applications of Mathematics, Tome 16 (1971) no. 3, pp. 203-214. doi: 10.21136/AM.1971.103346
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