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 Fibonacci

 http://functions.wolfram.com/07.06.13.0005.01

 Input Form

 Derivative[2][w][z] + ((3 g[z] Derivative[1][g][z])/(4 + g[z]^2) - (2 Derivative[1][h][z])/h[z] - Derivative[2][g][z]/Derivative[1][g][z]) Derivative[1][w][z] + (((1 - \[Nu]^2) Derivative[1][g][z]^2)/ (4 + g[z]^2) - (3 g[z] Derivative[1][g][z] Derivative[1][h][z])/ ((4 + g[z]^2) h[z]) + (2 Derivative[1][h][z]^2)/h[z]^2 + (Derivative[1][h][z] Derivative[2][g][z])/(h[z] Derivative[1][g][z]) - Derivative[2][h][z]/h[z]) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] Fibonacci[\[Nu], g[z]] + Subscript[c, 2] (h[z]/(4 + g[z]^2)^(1/4)) LegendreP[-(1/2) + \[Nu], 1/2, 2, (I g[z])/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["3", " ", RowBox[List["g", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List["h", "[", "z", "]"]]], "-", FractionBox[RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]], ")"]], RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], " ", "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[Nu]", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], "-", FractionBox[RowBox[List["3", " ", RowBox[List["g", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], " ", RowBox[List["h", "[", "z", "]"]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], RowBox[List["h", "[", "z", "]"]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["h", "[", "z", "]"]], RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", FractionBox[RowBox[List["h", "[", "z", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "4"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", FractionBox["1", "2"], ",", "2", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["g", "[", "z", "]"]]]], "2"]]], "]"]]]]]]]]]]]]

 MathML Form

 w ′′ ( z ) + ( 3 g ( z ) g ( z ) g ( z ) 2 + 4 - 2 h ( z ) h ( z ) - g ′′ ( z ) g ( z ) ) w ( z ) + ( ( 1 - ν 2 ) g ( z ) 2 g ( z ) 2 + 4 - 3 g ( z ) h ( z ) g ( z ) ( g ( z ) 2 + 4 ) h ( z ) + 2 h ( z ) 2 h ( z ) 2 + h ( z ) g ′′ ( z ) h ( z ) g ( z ) - h ′′ ( z ) h ( z ) ) w ( z ) 0 /; w ( z ) c 1 h ( z ) F TagBox["F", Fibonacci] ν ( g ( z ) ) + c 2 h ( z ) g ( z ) 2 + 4 4 P TagBox["P", LegendreP] ν - 1 2 1 2 ( 2 g ( z ) TagBox[RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["g", "(", "z", ")"]]]], HoldComplete[LegendreP, 2]] ) Condition z 2 w z 3 g z z g z g z 2 4 -1 -1 2 z h z h z -1 -1 z 2 g z z g z -1 z w z 1 -1 ν 2 z g z 2 g z 2 4 -1 -1 3 g z z h z z g z g z 2 4 h z -1 2 z h z 2 h z 2 -1 z h z z 2 g z h z z g z -1 -1 z 2 h z h z -1 w z 0 w z Subscript c 1 h z Fibonacci ν g z Subscript c 2 h z g z 2 4 1 4 -1 LegendreP ν -1 1 2 1 2 2 2 -1 g z [/itex]

 Rule Form

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

 2007-05-02