(-Derivative[2][Subscript[\[Psi], k]][x] + e^(2 x) Subscript[\[Psi], k][x] == k^2 Subscript[\[Psi], k][x]) \[DoubleLongLeftArrow] (Subscript[\[Psi], k][x] == (Sqrt[Sinh[Pi k]] BesselK[\[DoubleStruckI] k, e^x])/Pi /; k > 0)

Cell[BoxData[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Psi]", "k", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "x", "]"]]]], "+", RowBox[List[SuperscriptBox["e", RowBox[List["2", " ", "x"]]], " ", RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["k", "2"], " ", RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]]]]]], ")"]], "\[DoubleLongLeftArrow]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]], "\[Equal]", FractionBox[RowBox[List[SqrtBox[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "k"]], "]"]]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["\[DoubleStruckI]", " ", "k"]], ",", SuperscriptBox["e", "x"]]], "]"]]]], "\[Pi]"]]], "/;", RowBox[List["k", ">", "0"]]]], ")"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <msubsup> <mi> &#968; </mi> <mi> k </mi> <mo> &#8243; </mo> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10232; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mrow> <mfrac> <msqrt> <mrow> <mi> sinh </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mrow> <mi> &#120154; </mi> <mo> &#8290; </mo> <mtext> </mtext> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msup> <mi> e </mi> <mi> x </mi> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> DoubleLongLeftArrow </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> k </ci> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> e </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> k </ci> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> k </ci> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> k </ci> </apply> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> sinh </ci> <apply> <times /> <pi /> <ci> k </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselK </ci> <apply> <times /> <ci> &#120154; </ci> <ci> k </ci> </apply> <apply> <power /> <ci> e </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List[SuperscriptBox[SubscriptBox["\[Psi]", "k"], "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "x", "]"]]]], "+", RowBox[List[SuperscriptBox["e", RowBox[List["2", " ", "x"]]], " ", RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["k", "2"], " ", RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]]]]]], ")"]], "\[DoubleLongLeftArrow]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "x", "]"]], "\[Equal]", FractionBox[RowBox[List[SqrtBox[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "k"]], "]"]]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["\[DoubleStruckI]", " ", "k"]], ",", SuperscriptBox["e", "x"]]], "]"]]]], "\[Pi]"]]], "/;", RowBox[List["k", ">", "0"]]]], ")"]]]]]]

2001-10-29