(Derivative[1][w][x] + E^x w[x] == \[Alpha]) \[DoubleLongLeftArrow] (w == Subscript[c, 1]/E^E^x + (\[Alpha] ExpIntegralEi[E^x])/E^E^x)

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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> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mi> x </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10232; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mi> x </mi> </msup> </mrow> </msup> <mo> &#8290; </mo> <mi> &#945; </mi> <mo> &#8290; </mo> <mrow> <mi> Ei </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mi> x </mi> </msup> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mi> x </mi> </msup> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> DoubleLongLeftArrow </ci> <apply> <eq /> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> x </ci> </bvar> <apply> <ci> w </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> <apply> <ci> w </ci> <ci> x </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <eq /> <ci> w </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> </apply> </apply> <ci> &#945; </ci> <apply> <ci> ExpIntegralEi </ci> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29