Bessel function of the second kind: Differential equations (formula 03.03.13.0022)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

http://functions.wolfram.com/03.03.13.0022.01

Input Form

Derivative[2][w][z] - (-(Derivative[1][g][z]/g[z]) + (2 Derivative[1][h][z])/h[z] + Derivative[2][g][z]/Derivative[1][g][z]) Derivative[1][w][z] - ((-1 + \[Nu]^2/g[z]^2) Derivative[1][g][z]^2 + (Derivative[1][g][z] Derivative[1][h][z])/(g[z] h[z]) - (Derivative[1][h][z] Derivative[2][g][z])/(h[z] Derivative[1][g][z]) + (-2 Derivative[1][h][z]^2 + h[z] Derivative[2][h][z])/h[z]^2) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] BesselJ[\[Nu], g[z]] + Subscript[c, 2] h[z] BesselY[\[Nu], g[z]]

Standard Form

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MathML Form

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<annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> h </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> g </ci> <ci> z </ci> </apply> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> h </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02