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variants of this functions
KelvinKei




Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKei[z] > Differential equations > Ordinary linear differential equations and wronskians





http://functions.wolfram.com/03.15.13.0005.01









  


  








Input Form





g[z]^4 h[z]^4 Derivative[1][g][z]^3 Derivative[4][w][z] + 2 g[z]^3 h[z]^3 Derivative[1][g][z]^2 (-2 g[z] Derivative[1][g][z] Derivative[1][h][z] + h[z] (Derivative[1][g][z]^2 - 3 g[z] Derivative[2][g][z])) Derivative[3][w][z] + g[z]^2 h[z]^2 Derivative[1][g][z] (12 g[z]^2 Derivative[1][g][z]^2 Derivative[1][h][z]^2 - 6 g[z] h[z] Derivative[1][g][z] (Derivative[1][g][z]^2 Derivative[1][h][z] - 3 g[z] Derivative[1][h][z] Derivative[2][g][z] + g[z] Derivative[1][g][z] Derivative[2][h][z]) - h[z]^2 (Derivative[1][g][z]^4 + 6 g[z] Derivative[1][g][z]^2 Derivative[2][g][z] - 15 g[z]^2 Derivative[2][g][z]^2 + 4 g[z]^2 Derivative[1][g][z] Derivative[3][g][z])) Derivative[2][w][z] + g[z] h[z] (-24 g[z]^3 Derivative[1][g][z]^3 Derivative[1][h][z]^3 + 12 g[z]^2 h[z] Derivative[1][g][z]^2 Derivative[1][h][z] (Derivative[1][g][z]^2 Derivative[1][h][z] - 3 g[z] Derivative[1][h][z] Derivative[2][g][z] + 2 g[z] Derivative[1][g][z] Derivative[2][h][z]) + 2 g[z] h[z]^2 Derivative[1][g][z] (Derivative[1][g][z]^4 Derivative[1][h][z] - 15 g[z]^2 Derivative[1][h][z] Derivative[2][g][z]^2 - 3 g[z] Derivative[1][g][z]^3 Derivative[2][h][z] + g[z]^2 Derivative[1][g][z] (9 Derivative[2][g][z] Derivative[2][h][z] + 4 Derivative[1][h][z] Derivative[3][g][z]) - 2 g[z] Derivative[1][g][z]^2 (-3 Derivative[1][h][z] Derivative[2][g][z] + g[z] Derivative[3][h][z])) + h[z]^3 (Derivative[1][g][z]^6 + g[z] Derivative[1][g][z]^4 Derivative[2][g][z] - 15 g[z]^3 Derivative[2][g][z]^3 - 2 g[z]^2 Derivative[1][g][z]^3 Derivative[3][g][z] + 10 g[z]^3 Derivative[1][g][z] Derivative[2][g][z] Derivative[3][g][z] + g[z]^2 Derivative[1][g][z]^2 (6 Derivative[2][g][z]^2 - g[z] Derivative[4][g][z]))) Derivative[1][w][z] + (g[z]^4 h[z]^4 Derivative[1][g][z]^7 + g[z]^2 h[z]^2 Derivative[1][g][z] (-2 Derivative[1][h][z]^2 + h[z] Derivative[2][h][z]) (Derivative[1][g][z]^4 + 6 g[z] Derivative[1][g][z]^2 Derivative[2][g][z] - 15 g[z]^2 Derivative[2][g][z]^2 + 4 g[z]^2 Derivative[1][g][z] Derivative[3][g][z]) - 2 g[z]^3 h[z] Derivative[1][g][z]^2 (Derivative[1][g][z]^2 - 3 g[z] Derivative[2][g][z]) (6 Derivative[1][h][z]^3 - 6 h[z] Derivative[1][h][z] Derivative[2][h][z] + h[z]^2 Derivative[3][h][z]) - g[z] h[z]^3 Derivative[1][h][z] (Derivative[1][g][z]^6 + g[z] Derivative[1][g][z]^4 Derivative[2][g][z] - 15 g[z]^3 Derivative[2][g][z]^3 - 2 g[z]^2 Derivative[1][g][z]^3 Derivative[3][g][z] + 10 g[z]^3 Derivative[1][g][z] Derivative[2][g][z] Derivative[3][g][z] + g[z]^2 Derivative[1][g][z]^2 (6 Derivative[2][g][z]^2 - g[z] Derivative[4][g][z])) + g[z]^4 Derivative[1][g][z]^3 (24 Derivative[1][h][z]^4 - 36 h[z] Derivative[1][h][z]^2 Derivative[2][h][z] + 8 h[z]^2 Derivative[1][h][z] Derivative[3][h][z] + h[z]^2 (6 Derivative[2][h][z]^2 - h[z] Derivative[4][h][z]))) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] KelvinBer[g[z]] + Subscript[c, 2] h[z] KelvinBei[g[z]] + Subscript[c, 3] h[z] KelvinKer[g[z]] + Subscript[c, 4] h[z] KelvinKei[g[z]]








Standard Form





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MathML Form







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<mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;3&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;3&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <semantics> <mrow> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;4&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;3&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> h </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> h </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;3&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> g </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> 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Rule Form





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





2007-05-02