Ordinary Differential Equation with variable coffiecient












1












$begingroup$


I am studying ODE. I got stuck in following question which was given in my text book exercise.




Find solution of D.E. $y''+ye^x=0$




I can't recognize the type. As $y'$ term is not there so I can't make this variable coefficient to constant coefficient problem. So please help me.






ordinary-differential-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
    $endgroup$
    – Story123
    Jan 8 at 9:50












  • $begingroup$
    For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
    $endgroup$
    – LutzL
    Jan 8 at 10:05
















1












$begingroup$


I am studying ODE. I got stuck in following question which was given in my text book exercise.




Find solution of D.E. $y''+ye^x=0$




I can't recognize the type. As $y'$ term is not there so I can't make this variable coefficient to constant coefficient problem. So please help me.






ordinary-differential-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
    $endgroup$
    – Story123
    Jan 8 at 9:50












  • $begingroup$
    For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
    $endgroup$
    – LutzL
    Jan 8 at 10:05














1












1








1


1



$begingroup$


I am studying ODE. I got stuck in following question which was given in my text book exercise.




Find solution of D.E. $y''+ye^x=0$




I can't recognize the type. As $y'$ term is not there so I can't make this variable coefficient to constant coefficient problem. So please help me.






ordinary-differential-equations







share|cite|improve this question











$endgroup$




I am studying ODE. I got stuck in following question which was given in my text book exercise.




Find solution of D.E. $y''+ye^x=0$




I can't recognize the type. As $y'$ term is not there so I can't make this variable coefficient to constant coefficient problem. So please help me.






ordinary-differential-equations




ordinary-differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 8 at 10:00









LutzL

60.6k42057




60.6k42057










asked Jan 8 at 9:33









Somya SinghSomya Singh

84




84












  • $begingroup$
    We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
    $endgroup$
    – Story123
    Jan 8 at 9:50












  • $begingroup$
    For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
    $endgroup$
    – LutzL
    Jan 8 at 10:05


















  • $begingroup$
    We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
    $endgroup$
    – Story123
    Jan 8 at 9:50












  • $begingroup$
    For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
    $endgroup$
    – LutzL
    Jan 8 at 10:05
















$begingroup$
We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
$endgroup$
– Story123
Jan 8 at 9:50






$begingroup$
We know for any $x$ we have a ordinary point, so by Fuchs Theorem, we know all the solutions have a power series solution. So expand $y,y’,e^x$ as a power series and solve for the coefficients to get the answer. It’ll be in terms of special functions known as the Bessel functions. (I don’t think it’s possible to convert this problem into a constant coefficient problem)
$endgroup$
– Story123
Jan 8 at 9:50














$begingroup$
For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
$endgroup$
– LutzL
Jan 8 at 10:05




$begingroup$
For large $xgg1$ you get via WKB approximation $y(x)sim e^{-x/4}(Acos(2e^{x/2})+Bsin(2e^{x/2}))$. So you could try to compute the equation for $y(x)=f(2e^{x/2})$ and compare that to the equations for special functions.
$endgroup$
– LutzL
Jan 8 at 10:05










1 Answer
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$begingroup$

You have a second-order linear ordinary differential equation (ODE) here.



This particular ODE can be transformed to Bessel's differential equation by the definition of a new independent variable $z := 2 e^{x/2}$.



edit: Basically the same as LutzL's suggestion.






share|cite|improve this answer











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    1 Answer
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    $begingroup$

    You have a second-order linear ordinary differential equation (ODE) here.



    This particular ODE can be transformed to Bessel's differential equation by the definition of a new independent variable $z := 2 e^{x/2}$.



    edit: Basically the same as LutzL's suggestion.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      You have a second-order linear ordinary differential equation (ODE) here.



      This particular ODE can be transformed to Bessel's differential equation by the definition of a new independent variable $z := 2 e^{x/2}$.



      edit: Basically the same as LutzL's suggestion.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        You have a second-order linear ordinary differential equation (ODE) here.



        This particular ODE can be transformed to Bessel's differential equation by the definition of a new independent variable $z := 2 e^{x/2}$.



        edit: Basically the same as LutzL's suggestion.






        share|cite|improve this answer











        $endgroup$



        You have a second-order linear ordinary differential equation (ODE) here.



        This particular ODE can be transformed to Bessel's differential equation by the definition of a new independent variable $z := 2 e^{x/2}$.



        edit: Basically the same as LutzL's suggestion.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 8 at 10:17

























        answered Jan 8 at 10:07









        ChristophChristoph

        59616




        59616






























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