Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes

Panc, Vladimír

Applications of Mathematics, Tome 16 (1971) no. 3, pp. 203-214.

Résumé

The paper gives a comprehensive review of general solutions of the ordinary linear differential equation (1) ${\mathrm{\Delta }}^{2}w+2ϵ\mathrm{\Delta }w+w=0,\mathrm{\Delta }=d^2/d{\rho }^2+(1/\rho )(d/d\rho )$, 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 $ϵ=\pm 1$ the corresponding characteristic equation ${\lambda }^2-2ϵ\lambda +1=0$ 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) $[\mathrm{\Delta }+\lambda -(v/\rho {)}^2{]}^{2}w=0$ can be written in the form $w=A_1{J}_v(\rho sqrt\lambda )+A_2\rho J_{v+1}(\rho \sqrt{\lambda })+A_3{Y}_v(\rho \sqrt{\lambda })+A_4\rho Y_{v+1}(\rho \sqrt{\lambda })$ 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.

Détail

DOI : 10.21136/AM.1971.103346

Classification : 30D05, 34M99, 39B22, 39B32, 74Bxx