confluent differential equation












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I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:



$$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$



where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.



I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.










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$endgroup$

















    0












    $begingroup$


    I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:



    $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$



    where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.



    I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:



      $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$



      where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.



      I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.










      share|cite|improve this question











      $endgroup$




      I have a similar problem and I would appreciate any help in this matter. The DE that I am dealing with is:



      $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$



      where $a, b, c$ are constant and $f_z$ is the first derivative of $f(z)$ and $f_{zz}$ is the second derivative.



      I believe that this eq. must be related to the confleunt differential equation, but for some reasons I cannot find this form in the books. Can anyone give a reference books where I can find this equation and its solution? Many thanks.







      ordinary-differential-equations






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      edited Jan 10 at 20:26









      greelious

      472112




      472112










      asked Jan 10 at 18:58









      user630770user630770

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          0












          $begingroup$

          $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
          On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :



          Confluent : $zf''+(c-x)f'-af=0$



          hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$



          The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.



          I am afraid that you will have to use numerical methods of solving.






          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            In fact this relates to Heun's Confluent Equation.






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              2 Answers
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              2 Answers
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              0












              $begingroup$

              $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
              On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :



              Confluent : $zf''+(c-x)f'-af=0$



              hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$



              The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.



              I am afraid that you will have to use numerical methods of solving.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
                On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :



                Confluent : $zf''+(c-x)f'-af=0$



                hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$



                The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.



                I am afraid that you will have to use numerical methods of solving.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
                  On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :



                  Confluent : $zf''+(c-x)f'-af=0$



                  hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$



                  The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.



                  I am afraid that you will have to use numerical methods of solving.






                  share|cite|improve this answer









                  $endgroup$



                  $$z(z-1)f_{zz} + (az-1)f_z+ (b+cz)f=0$$
                  On the basic forms, the confluent hypergeometric and the hypergeometric differential equation have constant coefficient for the $f(x)$ term :



                  Confluent : $zf''+(c-x)f'-af=0$



                  hypergeometric : $z(1-z)f''+(c-(a+b+1)z)f'-abf=0$



                  The coefficient $(b+cz)$ complicates a lot the problem. This might be related to some Meijer-G functions (I don't confirm this). I am not surprized that you cannot find this ODE in the usual books.



                  I am afraid that you will have to use numerical methods of solving.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 11 at 11:50









                  JJacquelinJJacquelin

                  44.3k21854




                  44.3k21854























                      0












                      $begingroup$

                      In fact this relates to Heun's Confluent Equation.






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        In fact this relates to Heun's Confluent Equation.






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          In fact this relates to Heun's Confluent Equation.






                          share|cite|improve this answer









                          $endgroup$



                          In fact this relates to Heun's Confluent Equation.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Jan 11 at 12:36









                          doraemonpauldoraemonpaul

                          12.7k31660




                          12.7k31660






























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