Derivative[2][w][z] - a z^n w[z] == 0 /; w[z] == Sqrt[z] (Subscript[c, 1] BesselI[1/(2 + n), (2 Sqrt[a] z^((2 + n)/2))/(2 + n)] + Subscript[c, 2] BesselI[-(1/(2 + n)), (2 Sqrt[a] z^((2 + n)/2))/(2 + n)]) && !Element[1/(2 + n), Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <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 /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29