Solution 31 to problem l112o23

Expressions | Parameters | Relevance | Back to problem l112o23

Expressions

The solution is given through the following expressions:


      1                2        1      2   2
     ----*a12*a14*a5*a6 *b24 - ----*a14 *a5 *a6*b24
      16                        16
b42=------------------------------------------------
                             5
                          a12


         3            2        3     2
      - ---*a12*a14*a6 *b24 + ---*a14 *a5*a6*b24
         8                     8
b41=---------------------------------------------
                           4
                        a12


         3        2        9
      - ---*a12*a6 *b24 + ---*a14*a5*a6*b24
         4                 8
b40=----------------------------------------
                         3
                      a12


         3     2   2        3                       3      2   2
      - ---*a12 *a6 *b24 + ---*a12*a14*a5*a6*b24 + ----*a14 *a5 *b24
         8                  8                       32
b39=-----------------------------------------------------------------
                                     4
                                  a12


      3            2        3     2
     ---*a12*a14*a6 *b24 - ---*a14 *a5*a6*b24
      4                     4
b38=------------------------------------------
                        3
                     a12 *a5


      3        2        9
     ---*a12*a6 *b24 - ---*a14*a5*a6*b24
      2                 4
b37=-------------------------------------
                      2
                   a12 *a5


      3     2   2        3                       3     2   2
     ---*a12 *a6 *b24 - ---*a12*a14*a5*a6*b24 - ---*a14 *a5 *b24
      2                  2                       8
b36=-------------------------------------------------------------
                                  3
                               a12 *a5


      3        2        9
     ---*a12*a6 *b24 - ---*a14*a5*a6*b24
      4                 8
b35=-------------------------------------
                      2
                   a12 *a5


         3
      - ---*a6*b24
         4
b34=---------------
          a12


         3
      - ---*a6*b24
         4
b33=---------------
          a12


         3                3
      - ---*a12*a6*b24 - ---*a14*a5*b24
         2                4
b32=------------------------------------
                       2
                    a12


         3
      - ---*a14*a5*b24
         4
b31=-------------------
              2
           a12


         1            2        1     2
      - ---*a12*a14*a6 *b24 + ---*a14 *a5*a6*b24
         2                     2
b30=---------------------------------------------
                         2   2
                      a12 *a5


         3     2   2        3                       3     2   2
      - ---*a12 *a6 *b24 + ---*a12*a14*a5*a6*b24 + ---*a14 *a5 *b24
         2                  2                       8
b29=----------------------------------------------------------------
                                   2   2
                                a12 *a5


         3        2        9
      - ---*a12*a6 *b24 + ---*a14*a5*a6*b24
         2                 4
b28=----------------------------------------
                          2
                    a12*a5


      3
     ---*a14*b24
      2
b27=-------------
         a12


      3                3
     ---*a12*a6*b24 + ---*a14*a5*b24
      2                4
b26=---------------------------------
                 a12*a5


      3
     ---*a6*b24
      4
b25=------------
         a5


      1             2   2        1      2   3
     ----*a12*a14*a5 *a6 *b24 - ----*a14 *a5 *a6*b24
      16                         16
b23=-------------------------------------------------
                             6
                          a12


         3               2        3     2   2
      - ---*a12*a14*a5*a6 *b24 + ---*a14 *a5 *a6*b24
         8                        8
b22=-------------------------------------------------
                             5
                          a12


         3     2      2        9            2           3      2   3
      - ---*a12 *a5*a6 *b24 + ---*a12*a14*a5 *a6*b24 + ----*a14 *a5 *b24
         4                     8                        16
b21=---------------------------------------------------------------------
                                       5
                                    a12


      3            2        3     2
     ---*a12*a14*a6 *b24 - ---*a14 *a5*a6*b24
      4                     4
b20=------------------------------------------
                          4
                       a12


      3     2   2        9                       3     2   2
     ---*a12 *a6 *b24 - ---*a12*a14*a5*a6*b24 - ---*a14 *a5 *b24
      2                  4                       4
b19=-------------------------------------------------------------
                                   4
                                a12


      3     2   2        3                       3      2   2
     ---*a12 *a6 *b24 - ---*a12*a14*a5*a6*b24 - ----*a14 *a5 *b24
      8                  4                       32
b18=--------------------------------------------------------------
                                    4
                                 a12


         3                   3        2
      - ---*a12*a5*a6*b24 - ---*a14*a5 *b24
         2                   4
b17=----------------------------------------
                         3
                      a12


         3                   3        2
      - ---*a12*a5*a6*b24 - ---*a14*a5 *b24
         4                   4
b16=----------------------------------------
                         3
                      a12


         3
      - ---*a5*a6*b24
         4
b15=------------------
              2
           a12


         1            2        1     2
      - ---*a12*a14*a6 *b24 + ---*a14 *a5*a6*b24
         2                     2
b14=---------------------------------------------
                          3
                       a12 *a5


      3     2
     ---*a14 *a5*b24
      4
b13=-----------------
             3
          a12


      3                    3     2
     ---*a12*a14*a6*b24 + ---*a14 *a5*b24
      4                    8
b12=--------------------------------------
                        3
                     a12


      3
     ---*a14*a5*b24
      2
b11=----------------
             2
          a12


b10=0


     3                3
    ---*a12*a6*b24 + ---*a14*a5*b24
     2                4
b9=---------------------------------
                    2
                 a12


     3
    ---*a6*b24
     4
b8=------------
       a12


     3
    ---*a5*b24
     4
b7=------------
       a12


        3
     - ---*a5*b24
        4
b6=---------------
         a12


b5=0


        3     2   2        3                       3     2   2
     - ---*a12 *a6 *b24 + ---*a12*a14*a5*a6*b24 - ---*a14 *a5 *b24
        2                  2                       8
b4=----------------------------------------------------------------
                                  2   2
                               a12 *a5


b3=0


b2=0


    1
b1=---*b24
    4


         1        2    1
      - ---*a12*a6  + ---*a14*a5*a6
         4             4
a19=--------------------------------
                     2
                  a12


           2
     a12*a6  - a14*a5*a6
a18=---------------------
           a12*a5


a17= - a6


         1
      - ---*a14*a5
         2
a16=---------------
          a12


              2
      - a12*a6  + a14*a5*a6
a15=------------------------
                2
              a5


     a12*a6
a13=--------
       a5


         1           2    1        2
      - ---*a12*a5*a6  + ---*a14*a5 *a6
         4                4
a11=------------------------------------
                       3
                    a12


           2
     a12*a6  - a14*a5*a6
a10=---------------------
               2
            a12


                    1        2
     - a12*a5*a6 - ---*a14*a5
                    2
a9=----------------------------
                  2
               a12


             2
     - a12*a6  + a14*a5*a6
a8=------------------------
            a12*a5


    a14*a5
a7=--------
     a12


a6=a6


a4=0


a3=0


a2=0


a1=0

Parameters

Apart from the condition that they must not vanish to give a non-trivial solution and a non-singular solution with non-vanishing denominators, the following parameters are free:

 b24,a6,a5,a14,a12

Relevance for the application:

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

                      3                     3   2                     2   2
df(u,t)=(u *f(v,v)*a12 *a5*a6 + f(v,v  )*a12 *a5  - f(v,v )*f(v,v)*a12 *a5 *a6
          x                          2x                  x

             1                           3              2       2
          - ---*f(v,v )*f(v,v)*a12*a14*a5  + f(v,v )*a12 *a14*a5 *u
             2       x                            x

             1        3       2   2    1        3       3
          - ---*f(v,v) *a12*a5 *a6  + ---*f(v,v) *a14*a5 *a6
             4                         4

                  2    2      2           2           2
          + f(v,v) *a12 *a5*a6 *u - f(v,v) *a12*a14*a5 *a6*u

                      3   2  2             2            2      3
          - f(v,v)*a12 *a6 *u  + f(v,v)*a12 *a14*a5*a6*u )/(a12 *a5)

               3                  3   2    1                      3
df(v,t)=(u *a12 *a5*a6*v + v  *a12 *a5  - ---*v *f(v,v)*a12*a14*a5
          x                 2x             2   x

                  2       2                2   2
          + v *a12 *a14*a5 *u - f(v,v )*a12 *a5 *a6*v
             x                       x

             1        2       2   2      1        2       3
          - ---*f(v,v) *a12*a5 *a6 *v + ---*f(v,v) *a14*a5 *a6*v
             4                           4

                      2      2                        2             3   2  2
          + f(v,v)*a12 *a5*a6 *u*v - f(v,v)*a12*a14*a5 *a6*u*v - a12 *a6 *u *v

               2            2        2   2
          + a12 *a14*a5*a6*u *v)/(a12 *a5 )

          1         6   2        3                5   2
df(u,s)=(---*u  *a12 *a5 *b24 + ---*u  *f(v,v)*a12 *a5 *a6*b24
          4   3x                 4   2x

             3                5   2           3                4       3
          + ---*u *f(v,v )*a12 *a5 *a6*b24 + ---*u *f(v,v )*a12 *a14*a5 *b24
             2   x      x                     4   x      x

             3           2    4   2   2        3           2    3       3
          + ---*u *f(v,v) *a12 *a5 *a6 *b24 - ---*u *f(v,v) *a12 *a14*a5 *a6*b24
             8   x                             4   x

             3            2    2    2   4
          - ----*u *f(v,v) *a12 *a14 *a5 *b24
             32   x

             3               4       2
          + ---*u *f(v,v)*a12 *a14*a5 *a6*b24*u
             4   x

             3               3    2   3          3        6   2      2
          + ---*u *f(v,v)*a12 *a14 *a5 *b24*u - ---*u *a12 *a6 *b24*u
             8   x                               2   x

             3        5                2    3        4    2   2      2
          + ---*u *a12 *a14*a5*a6*b24*u  - ---*u *a12 *a14 *a5 *b24*u
             2   x                          8   x

             3               5   3        3                     4   3
          - ---*f(v ,v  )*a12 *a5 *b24 - ---*f(v ,v )*f(v,v)*a12 *a5 *a6*b24
             4     x  2x                  4     x  x

             3              5   3        3                     4   3
          + ---*f(v,v  )*a12 *a5 *b24 - ---*f(v,v  )*f(v,v)*a12 *a5 *a6*b24
             4       3x                  4       2x

             3                     3       4
          - ---*f(v,v  )*f(v,v)*a12 *a14*a5 *b24
             4       2x

             3              4       3          3         2    4   3
          + ---*f(v,v  )*a12 *a14*a5 *b24*u - ---*f(v,v ) *a12 *a5 *a6*b24
             2       2x                        2       x

             3         2    3       4        3                2    3   3   2
          - ---*f(v,v ) *a12 *a14*a5 *b24 - ---*f(v,v )*f(v,v) *a12 *a5 *a6 *b24
             4       x                       4       x

             9                2    2       4
          + ---*f(v,v )*f(v,v) *a12 *a14*a5 *a6*b24
             8       x

             3                 2        2   5
          + ----*f(v,v )*f(v,v) *a12*a14 *a5 *b24
             16       x

             3                    4   2   2
          + ---*f(v,v )*f(v,v)*a12 *a5 *a6 *b24*u
             2       x

             9                    3       3
          - ---*f(v,v )*f(v,v)*a12 *a14*a5 *a6*b24*u
             4       x

             3                    2    2   4
          - ---*f(v,v )*f(v,v)*a12 *a14 *a5 *b24*u
             4       x

             3             3    2   3      2    1         4           4   2
          + ---*f(v,v )*a12 *a14 *a5 *b24*u  + ----*f(v,v) *a12*a14*a5 *a6 *b24
             4       x                          16

             1         4    2   5           3        3    2       3   2
          - ----*f(v,v) *a14 *a5 *a6*b24 - ---*f(v,v) *a12 *a14*a5 *a6 *b24*u
             16                             8

             3        3        2   4
          + ---*f(v,v) *a12*a14 *a5 *a6*b24*u
             8

             3        2    3       2   2      2
          + ---*f(v,v) *a12 *a14*a5 *a6 *b24*u
             4

             3        2    2    2   3         2
          - ---*f(v,v) *a12 *a14 *a5 *a6*b24*u
             4

             1            4          2      3
          - ---*f(v,v)*a12 *a14*a5*a6 *b24*u
             2

             1            3    2   2         3      6   2
          + ---*f(v,v)*a12 *a14 *a5 *a6*b24*u )/(a12 *a5 )
             2

          3         5                3           5
df(v,s)=(---*u  *a12 *a5*a6*b24*v + ---*u *v *a12 *a5*a6*b24
          4   2x                     2   x  x

             3           4       2        3               4      2
          + ---*u *v *a12 *a14*a5 *b24 + ---*u *f(v,v)*a12 *a5*a6 *b24*v
             4   x  x                     4   x

             9               3       2             3        5   2
          - ---*u *f(v,v)*a12 *a14*a5 *a6*b24*v - ---*u *a12 *a6 *b24*u*v
             8   x                                 2   x

             9        4                            5   2
          + ---*u *a12 *a14*a5*a6*b24*u*v + v  *a12 *a5 *b24
             4   x                           3x

             3                3       3        3         4       2
          - ---*v  *f(v,v)*a12 *a14*a5 *b24 + ---*v  *a12 *a14*a5 *b24*u
             4   2x                            2   2x

             3                4   2           3                3       3
          - ---*v *f(v,v )*a12 *a5 *a6*b24 - ---*v *f(v,v )*a12 *a14*a5 *b24
             2   x      x                     4   x      x

             3           2    3   2   2        3           2    2       3
          - ---*v *f(v,v) *a12 *a5 *a6 *b24 + ---*v *f(v,v) *a12 *a14*a5 *a6*b24
             8   x                             8   x

             3            2        2   4        3               4      2
          + ----*v *f(v,v) *a12*a14 *a5 *b24 + ---*v *f(v,v)*a12 *a5*a6 *b24*u
             32   x                             2   x

             3               3       2
          - ---*v *f(v,v)*a12 *a14*a5 *a6*b24*u
             2   x

             3               2    2   3          3        5   2      2
          - ---*v *f(v,v)*a12 *a14 *a5 *b24*u - ---*v *a12 *a6 *b24*u
             8   x                               2   x

             3        4                2    3        3    2   2      2
          + ---*v *a12 *a14*a5*a6*b24*u  + ---*v *a12 *a14 *a5 *b24*u
             2   x                          8   x

             3              4   2             3              4   2
          - ---*f(v ,v )*a12 *a5 *a6*b24*v - ---*f(v,v  )*a12 *a5 *a6*b24*v
             4     x  x                       4       2x

             3                    3   2   2
          - ---*f(v,v )*f(v,v)*a12 *a5 *a6 *b24*v
             4       x

             9                    2       3
          + ---*f(v,v )*f(v,v)*a12 *a14*a5 *a6*b24*v
             8       x

             3             4      2
          + ---*f(v,v )*a12 *a5*a6 *b24*u*v
             2       x

             9             3       2
          - ---*f(v,v )*a12 *a14*a5 *a6*b24*u*v
             4       x

             1         3           3   2          1         3    2   4
          + ----*f(v,v) *a12*a14*a5 *a6 *b24*v - ----*f(v,v) *a14 *a5 *a6*b24*v
             16                                   16

             3        2    2       2   2
          - ---*f(v,v) *a12 *a14*a5 *a6 *b24*u*v
             8

             3        2        2   3
          + ---*f(v,v) *a12*a14 *a5 *a6*b24*u*v
             8

             3            3          2      2
          + ---*f(v,v)*a12 *a14*a5*a6 *b24*u *v
             4

             3            2    2   2         2      1     4       2      3
          - ---*f(v,v)*a12 *a14 *a5 *a6*b24*u *v - ---*a12 *a14*a6 *b24*u *v
             4                                      2

             1     3    2            3        5   2
          + ---*a12 *a14 *a5*a6*b24*u *v)/(a12 *a5 )
             2