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Re: Solving a DE using Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90481] Re: Solving a DE using Mathematica
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 10 Jul 2008 06:38:08 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g51uce$6tj$1@smc.vnet.net>

Greg wrote:

> I'm having problems solving this problem although it should appear pretty straightfoward:
> 
> (-l^2 - m^2 + n^2/(r + 0.016 z)^2) Z[z] + Z''[z] == 0
> 
> I am solving for Z[z].  These are the lines I use:
> 
> DSolve[Above Equation, Z[z], z]
> 
> I get an odd solution so I do a solution check plugging back in Z[z] and Z''[z] yet I don't get 0.  In the above, l,m,n,r are all constants.

Since we are looking for an analytic solution, we use using exact 
coefficients rather than machine-size numbers.

Also, we ask for a pure function, which is easier to manipulate and plug 
back into the original DE.

Finally, we check the result with *FullSimplify* since the solution 
involve special functions.

     de = (-l^2 - m^2 + n^2/(r + 0.016 z)^2) Z[z] + Z''[z] == 0

     de = Rationalize[de]

     sol = DSolve[de, Z, z]

     de /. sol[[1]]

     FullSimplify[%]


Regards,
-- Jean-Marc

In[1]:= de = (-l^2 - m^2 + n^2/(r + 0.016 z)^2) Z[z] + Z''[z] == 0

Out[1]=
                    2
    2    2         n
(-l  - m  + --------------) Z[z] + Z''[z] == 0
                          2
             (r + 0.016 z)

In[2]:= de = Rationalize[de]

Out[2]=
                  2
    2    2       n
(-l  - m  + ----------) Z[z] + Z''[z] == 0
                  2 z 2
             (r + ---)
                  125

In[3]:= sol = DSolve[de, Z, z]

Out[3]=
                                           1                     2
{{Z -> Function[{z}, C[1] WhittakerM[0, -(-) I Sqrt[-1 + 15625 n ],
                                           2

                   2    2              2    2
         125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] +

                            1                     2
       C[2] WhittakerW[0, -(-) I Sqrt[-1 + 15625 n ],
                            2

                   2    2              2    2
         125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z]]}}

In[4]:= de /. sol[[1]]

Out[4]=
     2    2            1   1                    2
4 (l  + m ) C[1] (-(((- - - I Sqrt[-1 + 15625 n ])
                       2   2

                            1                     2
            WhittakerM[1, -(-) I Sqrt[-1 + 15625 n ],
                            2

                       2    2              2    2
             125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z]) /

                     2    2              2    2    2
          (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z) ) +

       1  1                 1                     2
       - (- WhittakerM[0, -(-) I Sqrt[-1 + 15625 n ],
       2  2                 2

                      2    2              2    2
            125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] +

            1   1                    2
          ((- - - I Sqrt[-1 + 15625 n ])
            2   2

                             1                     2
             WhittakerM[1, -(-) I Sqrt[-1 + 15625 n ],
                             2

                        2    2              2    2
              125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z]) /

                      2    2              2    2
           (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)) +

         1   1                    2
       ((- - - I Sqrt[-1 + 15625 n ])
         2   2

            1                      1
          ((- - ---------------------------------------)
            2             2    2              2    2
                125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z

                             1                     2
             WhittakerM[1, -(-) I Sqrt[-1 + 15625 n ],
                             2

                        2    2              2    2
              125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] +

              3   1                    2
            ((- - - I Sqrt[-1 + 15625 n ])
              2   2

                               1                     2
               WhittakerM[2, -(-) I Sqrt[-1 + 15625 n ],
                               2

                          2    2              2    2
                125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z]) /

                        2    2              2    2
             (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z))) /

                   2    2              2    2
        (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)) +

                     2
       2    2       n
    (-l  - m  + ----------) (C[1]
                     2 z 2
                (r + ---)
                     125

                        1                     2
        WhittakerM[0, -(-) I Sqrt[-1 + 15625 n ],
                        2

                   2    2              2    2
         125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] +

                            1                     2
       C[2] WhittakerW[0, -(-) I Sqrt[-1 + 15625 n ],
                            2

                   2    2              2    2
         125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z]) +

        2    2                        1                     2
    4 (l  + m ) C[2] (WhittakerW[1, -(-) I Sqrt[-1 + 15625 n ],
                                      2

                   2    2              2    2
         125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] /

                   2    2              2    2    2
        (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)  +

       1  1                 1                     2
       - (- WhittakerW[0, -(-) I Sqrt[-1 + 15625 n ],
       2  2                 2

                      2    2              2    2
            125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] -

                          1                     2
          WhittakerW[1, -(-) I Sqrt[-1 + 15625 n ],
                          2

                      2    2              2    2
            125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] /

                      2    2              2    2
           (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)) -

         1                      1
       ((- - ---------------------------------------)
         2             2    2              2    2
             125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z

                           1                     2
           WhittakerW[1, -(-) I Sqrt[-1 + 15625 n ],
                           2

                      2    2              2    2
            125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] -

                          1                     2
          WhittakerW[2, -(-) I Sqrt[-1 + 15625 n ],
                          2

                      2    2              2    2
            125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z] /

                      2    2              2    2
           (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)) /

                   2    2              2    2
        (125 Sqrt[l  + m ] r + 2 Sqrt[l  + m ] z)) == 0

In[5]:= FullSimplify[%]

Out[5]= True


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