Solution 2 to problem l112o23

Expressions | Parameters | Relevance | Back to problem l112o23

Expressions

The solution is given through the following expressions:


         1      2       3        1      3   3
      - ----*a13 *a14*a4 *b24 + ----*a14 *a4 *b24
         64                      64
b42=----------------------------------------------
                            6
                         a12


         3      2       2        3      3   2
      - ----*a13 *a14*a4 *b24 + ----*a14 *a4 *b24
         32                      32
b41=----------------------------------------------
                            5
                         a12


      3     2   2        3     2   2
     ---*a13 *a4 *b24 + ---*a14 *a4 *b24
      8                  8
b40=-------------------------------------
                       4
                    a12


      3      2   2
     ----*a14 *a4 *b24
      16
b39=-------------------
              4
           a12


         3      2               3      3
      - ----*a13 *a14*a4*b24 + ----*a14 *a4*b24
         16                     16
b38=--------------------------------------------
                           4
                        a12


      3     2           3     2
     ---*a13 *a4*b24 + ---*a14 *a4*b24
      4                 4
b37=-----------------------------------
                      3
                   a12


      3     2
     ---*a14 *a4*b24
      4
b36=-----------------
             3
          a12


      3     2           3     2
     ---*a13 *a4*b24 + ---*a14 *a4*b24
      8                 8
b35=-----------------------------------
                      3
                   a12


      3
     ---*a13*a4*b24
      2
b34=----------------
             2
          a12


      3
     ---*a13*a4*b24
      2
b33=----------------
             2
          a12


      3                3
     ---*a13*a4*b24 + ---*a14*a4*b24
      2                2
b32=---------------------------------
                     2
                  a12


      3
     ---*a14*a4*b24
      4
b31=----------------
             2
          a12


         1     2            1     3
      - ---*a13 *a14*b24 + ---*a14 *b24
         8                  8
b30=------------------------------------
                       3
                    a12


      3     2
     ---*a14 *b24
      4
b29=--------------
            2
         a12


      3     2        3     2
     ---*a13 *b24 + ---*a14 *b24
      4              4
b28=-----------------------------
                   2
                a12


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


      3             3
     ---*a13*b24 + ---*a14*b24
      2             2
b26=---------------------------
                a12


      3
     ---*a13*b24
      2
b25=-------------
         a12


      1      2       4        1      3   4
     ----*a13 *a14*a4 *b24 - ----*a14 *a4 *b24
      64                      64
b23=-------------------------------------------
                          7
                       a12


      3      2       3        3      3   3
     ----*a13 *a14*a4 *b24 - ----*a14 *a4 *b24
      32                      32
b22=-------------------------------------------
                          6
                       a12


         3      2   3        9      2   3
      - ----*a13 *a4 *b24 - ----*a14 *a4 *b24
         16                  16
b21=------------------------------------------
                          5
                       a12


      3      2       2        3      3   2
     ----*a13 *a14*a4 *b24 - ----*a14 *a4 *b24
      16                      16
b20=-------------------------------------------
                          5
                       a12


         3     2   2
      - ---*a14 *a4 *b24
         2
b19=---------------------
               4
            a12


         3      2   2        3     2   2
      - ----*a13 *a4 *b24 - ---*a14 *a4 *b24
         16                  8
b18=-----------------------------------------
                         4
                      a12


         3        2
      - ---*a14*a4 *b24
         2
b17=--------------------
               3
            a12


         3        2        3        2
      - ---*a13*a4 *b24 - ---*a14*a4 *b24
         4                 4
b16=--------------------------------------
                        3
                     a12


         3        2
      - ---*a13*a4 *b24
         4
b15=--------------------
               3
            a12


      1     2               1     3
     ---*a13 *a14*a4*b24 - ---*a14 *a4*b24
      8                     8
b14=---------------------------------------
                        4
                     a12


      3     2           3     2
     ---*a13 *a4*b24 - ---*a14 *a4*b24
      4                 4
b13=-----------------------------------
                      3
                   a12


         3     2
      - ---*a14 *a4*b24
         4
b12=--------------------
               3
            a12


      3                3
     ---*a13*a4*b24 - ---*a14*a4*b24
      2                2
b11=---------------------------------
                     2
                  a12


      3
     ---*a13*a4*b24
      2
b10=----------------
             2
          a12


     3                3
    ---*a13*a4*b24 - ---*a14*a4*b24
     2                2
b9=---------------------------------
                    2
                 a12


        3
     - ---*a13*a4*b24
        4
b8=-------------------
             2
          a12


b7=0


    3*a4*b24
b6=----------
      a12


b5=0


     3     2
    ---*a13 *b24
     4
b4=--------------
           2
        a12


     3
    ---*a13*b24
     2
b3=-------------
        a12


     3
    ---*a13*b24
     2
b2=-------------
        a12


b1=b24


         1             2    1      2   2
      - ----*a13*a14*a4  + ----*a14 *a4
         16                 16
a19=-------------------------------------
                       3
                    a12


         1                1     2
      - ---*a13*a14*a4 + ---*a14 *a4
         4                4
a18=---------------------------------
                     2
                  a12


     a13*a4
a17=--------
      a12


      1
     ---*a14*a4
      2
a16=------------
        a12


         1             1     2
      - ---*a13*a14 + ---*a14
         4             4
a15=---------------------------
                a12


      1             3    1      2   3
     ----*a13*a14*a4  - ----*a14 *a4
      16                 16
a11=----------------------------------
                      4
                   a12


      1            2    1     2   2
     ---*a13*a14*a4  - ---*a14 *a4
      4                 4
a10=--------------------------------
                     3
                  a12


        1        2    1        2
     - ---*a13*a4  - ---*a14*a4
        2             2
a9=------------------------------
                   2
                a12


     1                1     2
    ---*a13*a14*a4 - ---*a14 *a4
     4                4
a8=------------------------------
                   2
                a12


    a13*a4 - a14*a4
a7=-----------------
          a12


        1
     - ---*a13*a4
        2
a6=---------------
         a12


a5=0


a3=0


a2=a13


a1=a12

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,a13,a4,a14,a12

Relevance for the application:

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

                5    1               3                4                     4
df(u,t)=(u  *a12  - ---*u *f(v,v)*a12 *a13*a4 + u *a12 *a13*u + f(v ,v )*a12 *a4
          2x         2   x                       x                 x  x

             1                    2       2    1                    2       2
          - ---*f(v,v )*f(v,v)*a12 *a13*a4  - ---*f(v,v )*f(v,v)*a12 *a14*a4
             2       x                         2       x

                       3                       3
          + f(v,v )*a12 *a13*a4*u - f(v,v )*a12 *a14*a4*u
                 x                       x

             1         3           3    1         3    2   3
          + ----*f(v,v) *a13*a14*a4  - ----*f(v,v) *a14 *a4
             16                         16

             1        2               2      1        2        2   2
          + ---*f(v,v) *a12*a13*a14*a4 *u - ---*f(v,v) *a12*a14 *a4 *u
             4                               4

             1            2             2    1            2    2     2     4
          + ---*f(v,v)*a12 *a13*a14*a4*u  - ---*f(v,v)*a12 *a14 *a4*u )/a12
             4                               4

               3                4    1               2                3
df(v,t)=(u *a12 *a13*v + v  *a12  + ---*v *f(v,v)*a12 *a14*a4 + v *a12 *a14*u
          x               2x         2   x                       x

                       2             1         2           2
          + f(v,v )*a12 *a13*a4*v - ----*f(v,v) *a13*a14*a4 *v
                 x                   16

             1         2    2   2      1
          + ----*f(v,v) *a14 *a4 *v - ---*f(v,v)*a12*a13*a14*a4*u*v
             16                        4

             1                2           1     2          2
          + ---*f(v,v)*a12*a14 *a4*u*v - ---*a12 *a13*a14*u *v
             4                            4

             1     2    2  2       3
          + ---*a12 *a14 *u *v)/a12
             4

                7        3                5               3         6
df(u,s)=(u  *a12 *b24 - ---*u  *f(v,v)*a12 *a13*a4*b24 + ---*u  *a12 *a13*b24*u
          3x             4   2x                           2   2x

             3    2    6            3                5
          + ---*u  *a12 *a13*b24 + ---*u *f(v,v )*a12 *a13*a4*b24
             2   x                  2   x      x

             3                5               3            2    3    2   2
          - ---*u *f(v,v )*a12 *a14*a4*b24 - ----*u *f(v,v) *a12 *a13 *a4 *b24
             2   x      x                     16   x

             3           2    3    2   2        3               4    2
          - ---*u *f(v,v) *a12 *a14 *a4 *b24 - ---*u *f(v,v)*a12 *a14 *a4*b24*u
             8   x                              4   x

             3        5    2      2                  6
          + ---*u *a12 *a13 *b24*u  + 3*f(v ,v  )*a12 *a4*b24
             4   x                         x  2x

             3                     4       2
          - ---*f(v ,v )*f(v,v)*a12 *a13*a4 *b24
             4     x  x

             3              5
          + ---*f(v ,v )*a12 *a13*a4*b24*u
             2     x  x

             3                     4       2
          - ---*f(v,v  )*f(v,v)*a12 *a13*a4 *b24
             4       2x

             3                     4       2
          - ---*f(v,v  )*f(v,v)*a12 *a14*a4 *b24
             4       2x

             3              5                 3              5
          + ---*f(v,v  )*a12 *a13*a4*b24*u - ---*f(v,v  )*a12 *a14*a4*b24*u
             2       2x                       2       2x

             3         2    4       2
          - ---*f(v,v ) *a12 *a14*a4 *b24
             2       x

             3                 2    2    2   3
          - ----*f(v,v )*f(v,v) *a12 *a13 *a4 *b24
             16       x

             9                 2    2    2   3
          - ----*f(v,v )*f(v,v) *a12 *a14 *a4 *b24
             16       x

             3                    3    2   2
          - ---*f(v,v )*f(v,v)*a12 *a14 *a4 *b24*u
             2       x

             3             4    2         2    3             4    2         2
          + ---*f(v,v )*a12 *a13 *a4*b24*u  - ---*f(v,v )*a12 *a14 *a4*b24*u
             4       x                         4       x

             1         4    2       4        1         4    3   4
          + ----*f(v,v) *a13 *a14*a4 *b24 - ----*f(v,v) *a14 *a4 *b24
             64                              64

             3         3        2       3
          + ----*f(v,v) *a12*a13 *a14*a4 *b24*u
             32

             3         3        3   3
          - ----*f(v,v) *a12*a14 *a4 *b24*u
             32

             3         2    2    2       2      2
          + ----*f(v,v) *a12 *a13 *a14*a4 *b24*u
             16

             3         2    2    3   2      2
          - ----*f(v,v) *a12 *a14 *a4 *b24*u
             16

             1            3    2             3    1            3    3         3
          + ---*f(v,v)*a12 *a13 *a14*a4*b24*u  - ---*f(v,v)*a12 *a14 *a4*b24*u )
             8                                    8

    7
/a12

          3         5              3           5
df(v,s)=(---*u  *a12 *a13*b24*v + ---*u *v *a12 *a13*b24
          2   2x                   2   x  x

             3           5            3               3    2
          + ---*u *v *a12 *a14*b24 + ---*u *f(v,v)*a12 *a13 *a4*b24*v
             2   x  x                 8   x

             3               3    2             3        4    2
          + ---*u *f(v,v)*a12 *a14 *a4*b24*v + ---*u *a12 *a13 *b24*u*v
             8   x                              4   x

             3        4    2                  6
          + ---*u *a12 *a14 *b24*u*v + v  *a12 *b24
             4   x                      3x

             3                4               3         5
          + ---*v  *f(v,v)*a12 *a14*a4*b24 + ---*v  *a12 *a14*b24*u
             4   2x                           2   2x

             3                4               3                4
          + ---*v *f(v,v )*a12 *a13*a4*b24 + ---*v *f(v,v )*a12 *a14*a4*b24
             2   x      x                     2   x      x

             3            2    2    2   2        3               3    2
          + ----*v *f(v,v) *a12 *a14 *a4 *b24 + ---*v *f(v,v)*a12 *a14 *a4*b24*u
             16   x                              4   x

             3        4    2      2    3              4
          + ---*v *a12 *a14 *b24*u  + ---*f(v ,v )*a12 *a13*a4*b24*v
             4   x                     2     x  x

             3              4
          + ---*f(v,v  )*a12 *a13*a4*b24*v
             2       2x

             3                    2    2   2
          + ---*f(v,v )*f(v,v)*a12 *a13 *a4 *b24*v
             8       x

             3                    2    2   2
          + ---*f(v,v )*f(v,v)*a12 *a14 *a4 *b24*v
             8       x

             3             3    2               3             3    2
          + ---*f(v,v )*a12 *a13 *a4*b24*u*v + ---*f(v,v )*a12 *a14 *a4*b24*u*v
             4       x                          4       x

             1         3    2       3          1         3    3   3
          - ----*f(v,v) *a13 *a14*a4 *b24*v + ----*f(v,v) *a14 *a4 *b24*v
             64                                64

             3         2        2       2
          - ----*f(v,v) *a12*a13 *a14*a4 *b24*u*v
             32

             3         2        3   2
          + ----*f(v,v) *a12*a14 *a4 *b24*u*v
             32

             3             2    2             2
          - ----*f(v,v)*a12 *a13 *a14*a4*b24*u *v
             16

             3             2    3         2      1     3    2          3
          + ----*f(v,v)*a12 *a14 *a4*b24*u *v - ---*a12 *a13 *a14*b24*u *v
             16                                  8

             1     3    3      3       6
          + ---*a12 *a14 *b24*u *v)/a12
             8