Solution 5 to problem l112o13

Expressions | Parameters | Relevance | Back to problem l112o13

Expressions

The solution is given through the following expressions:


      1     2   3          1       3   2        1    3   3
     ----*a0 *a3 *a7*b4 + ---*a0*a3 *a7 *b25 - ---*a3 *a7 *b1
      24                   8                    8
b42=----------------------------------------------------------
                                 6
                               a0


      1    2   2          3       2   2        3    2   3
     ---*a0 *a3 *a7*b4 + ---*a0*a3 *a7 *b25 - ---*a3 *a7 *b1
      4                   4                    4
b41=---------------------------------------------------------
                                 5
                               a0


      1       2          3       2           3    2   2
     ---*a0*a3 *a7*b2 - ---*a0*a3 *a7*b25 + ---*a3 *a7 *b1
      2                  2                   2
b40=-------------------------------------------------------
                                4
                              a0


         3       2           3    2   2
      - ---*a0*a3 *a7*b25 + ---*a3 *a7 *b1
         4                   4
b39=---------------------------------------
                        4
                      a0


      1    2             3          2        3       3
     ---*a0 *a3*a7*b4 + ---*a0*a3*a7 *b25 - ---*a3*a7 *b1
      2                  2                   2
b38=------------------------------------------------------
                               4
                             a0


                                           2
     a0*a3*a7*b2 - 3*a0*a3*a7*b25 + 3*a3*a7 *b1
b37=--------------------------------------------
                          3
                        a0


                                2
      - 3*a0*a3*a7*b25 + 3*a3*a7 *b1
b36=---------------------------------
                     3
                   a0


      1                 3                  3       2
     ---*a0*a3*a7*b2 - ---*a0*a3*a7*b25 + ---*a3*a7 *b1
      2                 2                  2
b35=----------------------------------------------------
                              3
                            a0


     a3*b25
b34=--------
       a0


     a3*b25
b33=--------
       a0


     3*a0*a3*b25 - 3*a3*a7*b1
b32=--------------------------
                 2
               a0


      3               3
     ---*a0*a3*b25 - ---*a3*a7*b1
      2               2
b31=------------------------------
                   2
                 a0


      1    2              2         3
     ---*a0 *a7*b4 + a0*a7 *b25 - a7 *b1
      3
b30=-------------------------------------
                       3
                     a0


                          2
      - 3*a0*a7*b25 + 3*a7 *b1
b29=---------------------------
                  2
                a0


                                  2
     a0*a7*b2 - 3*a0*a7*b25 + 3*a7 *b1
b28=-----------------------------------
                      2
                    a0


     3*a0*b25 - 3*a7*b1
b27=--------------------
             a0


     3*a0*b25 - 3*a7*b1
b26=--------------------
             a0


      - a0*b25 + a7*b1
b24=-------------------
            a7


         1     2   4          1       4   2        1    4   3
      - ----*a0 *a3 *a7*b4 - ---*a0*a3 *a7 *b25 + ---*a3 *a7 *b1
         24                   8                    8
b23=-------------------------------------------------------------
                                   7
                                 a0


         1    2   3          3       3   2        3    3   3
      - ---*a0 *a3 *a7*b4 - ---*a0*a3 *a7 *b25 + ---*a3 *a7 *b1
         4                   4                    4
b22=------------------------------------------------------------
                                  6
                                a0


      1    2   3       1       3          9       3           9    3   2
     ---*a0 *a3 *b4 - ---*a0*a3 *a7*b2 + ---*a0*a3 *a7*b25 - ---*a3 *a7 *b1
      4                2                  4                   4
b21=------------------------------------------------------------------------
                                        5
                                      a0


         1    2   2          3       2   2        3    2   3
      - ---*a0 *a3 *a7*b4 - ---*a0*a3 *a7 *b25 + ---*a3 *a7 *b1
         2                   2                    2
b20=------------------------------------------------------------
                                  5
                                a0


       2   2           2                2              2   2
     a0 *a3 *b4 - a0*a3 *a7*b2 + 6*a0*a3 *a7*b25 - 6*a3 *a7 *b1
b19=------------------------------------------------------------
                                  4
                                a0


      1    2   2       1       2          3       2           3    2   2
     ---*a0 *a3 *b4 - ---*a0*a3 *a7*b2 + ---*a0*a3 *a7*b25 - ---*a3 *a7 *b1
      4                2                  2                   2
b18=------------------------------------------------------------------------
                                        4
                                      a0


          2             2           2
     a0*a3 *b2 - 3*a0*a3 *b25 + 3*a3 *a7*b1
b17=----------------------------------------
                        3
                      a0


      1       2       5       2        3    2
     ---*a0*a3 *b2 - ---*a0*a3 *b25 + ---*a3 *a7*b1
      2               2                2
b16=------------------------------------------------
                            3
                          a0


      1    2        2
     ---*a3 *b2 - a3 *b25
      2
b15=----------------------
               2
             a0


         1    2                    2            3
      - ---*a0 *a3*a7*b4 - a0*a3*a7 *b25 + a3*a7 *b1
         3
b14=-------------------------------------------------
                             4
                           a0


       2                                 2
     a0 *a3*b4 + 3*a0*a3*a7*b25 - 3*a3*a7 *b1
b13=------------------------------------------
                         3
                       a0


       2                                               2
     a0 *a3*b4 - a0*a3*a7*b2 + 3*a0*a3*a7*b25 - 3*a3*a7 *b1
b12=--------------------------------------------------------
                                3
                              a0


     a0*a3*b2 - 3*a0*a3*b25 + 3*a3*a7*b1
b11=-------------------------------------
                       2
                     a0


     a3*b2
b10=-------
      a0


    2*a0*a3*b2 - 3*a0*a3*b25 + 3*a3*a7*b1
b9=---------------------------------------
                       2
                     a0


     1
    ---*a3*b2 - a3*b25
     2
b8=--------------------
            a0


    a3*b25
b7=--------
      a7


    3*a3*b1
b6=---------
      a0


b5=0


b3=b2


     1
    ---*a3*a7
     2
a8=-----------
       a0


        1    2
     - ---*a3 *a7
        2
a5=---------------
           2
         a0


     - a3*a7
a4=----------
       a0


a2=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:

 b25, b4, b1, b2, a3, a0, a7

Relevance for the application:

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

df(u,t)=

      3        2                2       1        2   2
 u *a0  + u *a0 *a6 + f(v,v )*a0 *a3 - ---*f(v,v) *a3 *a7 - f(v,v)*a0*a3*a7*u
  x        x               x            2
------------------------------------------------------------------------------
                                       2
                                     a0

                     1
         v *a0*a6 + ---*f(v,v)*a3*a7*v + a0*a7*u*v
          x          2
df(v,t)=-------------------------------------------
                            a0

               7          1               6                         6
df(u,s)=(u  *a0 *a7*b1 + ---*u  *f(v,v)*a0 *a3*a7*b2 - u  *f(v,v)*a0 *a3*a7*b25
          3x              2   2x                        2x

                  7             2   7                        6
          + u  *a0 *a7*b2*u + u  *a0 *a7*b2 + 2*u *f(v,v )*a0 *a3*a7*b2
             2x                x                 x      x

                           6                            5      2
          - 3*u *f(v,v )*a0 *a3*a7*b25 + 3*u *f(v,v )*a0 *a3*a7 *b1
               x      x                     x      x

             1           2   5   2          1           2   4   2   2
          + ---*u *f(v,v) *a0 *a3 *a7*b4 - ---*u *f(v,v) *a0 *a3 *a7 *b2
             4   x                          2   x

             3           2   4   2   2        3           2   3   2   3
          + ---*u *f(v,v) *a0 *a3 *a7 *b25 - ---*u *f(v,v) *a0 *a3 *a7 *b1
             2   x                            2   x

                        6                          5      2
          + u *f(v,v)*a0 *a3*a7*b4*u - u *f(v,v)*a0 *a3*a7 *b2*u
             x                          x

                          5      2                       4      3
          + 3*u *f(v,v)*a0 *a3*a7 *b25*u - 3*u *f(v,v)*a0 *a3*a7 *b1*u
               x                              x

                 7        2                 6
          + u *a0 *a7*b4*u  + 3*f(v ,v  )*a0 *a3*a7*b1
             x                     x  2x

             1                    5   2                           5   2
          + ---*f(v ,v )*f(v,v)*a0 *a3 *a7*b2 - f(v ,v )*f(v,v)*a0 *a3 *a7*b25
             2     x  x                            x  x

                       6                         7
          + f(v ,v )*a0 *a3*a7*b2*u + f(v,v  )*a0 *a3*b25
               x  x                        3x

             1                    5   2
          + ---*f(v,v  )*f(v,v)*a0 *a3 *a7*b2
             2       2x

             5                    5   2
          - ---*f(v,v  )*f(v,v)*a0 *a3 *a7*b25
             2       2x

             3                    4   2   2                 6
          + ---*f(v,v  )*f(v,v)*a0 *a3 *a7 *b1 + f(v,v  )*a0 *a3*a7*b2*u
             2       2x                               2x

                         6                            5      2
          - 3*f(v,v  )*a0 *a3*a7*b25*u + 3*f(v,v  )*a0 *a3*a7 *b1*u
                   2x                           2x

                   2   5   2                  2   5   2
          + f(v,v ) *a0 *a3 *a7*b2 - 3*f(v,v ) *a0 *a3 *a7*b25
                 x                          x

                     2   4   2   2       1                2   4   3
          + 3*f(v,v ) *a0 *a3 *a7 *b1 + ---*f(v,v )*f(v,v) *a0 *a3 *a7*b4
                   x                     4       x

             1                2   3   3   2
          - ---*f(v,v )*f(v,v) *a0 *a3 *a7 *b2
             2       x

             9                2   3   3   2
          + ---*f(v,v )*f(v,v) *a0 *a3 *a7 *b25
             4       x

             9                2   2   3   3                       5   2
          - ---*f(v,v )*f(v,v) *a0 *a3 *a7 *b1 + f(v,v )*f(v,v)*a0 *a3 *a7*b4*u
             4       x                                x

                             4   2   2
          - f(v,v )*f(v,v)*a0 *a3 *a7 *b2*u
                 x

                               4   2   2
          + 6*f(v,v )*f(v,v)*a0 *a3 *a7 *b25*u
                   x

                               3   2   3                  6           2
          - 6*f(v,v )*f(v,v)*a0 *a3 *a7 *b1*u + f(v,v )*a0 *a3*a7*b4*u
                   x                                 x

                        5      2      2               4      3     2
          + 3*f(v,v )*a0 *a3*a7 *b25*u  - 3*f(v,v )*a0 *a3*a7 *b1*u
                   x                             x

             1         4   2   4   2       1        4      4   3
          - ----*f(v,v) *a0 *a3 *a7 *b4 - ---*f(v,v) *a0*a3 *a7 *b25
             24                            8

             1        4   4   4       1        3   3   3   2
          + ---*f(v,v) *a3 *a7 *b1 - ---*f(v,v) *a0 *a3 *a7 *b4*u
             8                        4

             3        3   2   3   3          3        3      3   4
          - ---*f(v,v) *a0 *a3 *a7 *b25*u + ---*f(v,v) *a0*a3 *a7 *b1*u
             4                               4

             1        2   4   2   2     2    3        2   3   2   3      2
          - ---*f(v,v) *a0 *a3 *a7 *b4*u  - ---*f(v,v) *a0 *a3 *a7 *b25*u
             2                               2

             3        2   2   2   4     2    1           5      2     3
          + ---*f(v,v) *a0 *a3 *a7 *b1*u  - ---*f(v,v)*a0 *a3*a7 *b4*u
             2                               3

                     4      3      3            3      4     3     7
          - f(v,v)*a0 *a3*a7 *b25*u  + f(v,v)*a0 *a3*a7 *b1*u )/(a0 *a7)

               6                      6                    5   2
df(v,s)=(u  *a0 *a7*b25*v + 3*u *v *a0 *a7*b25 - 3*u *v *a0 *a7 *b1
          2x                   x  x                 x  x

             1              4      2         3              4      2
          + ---*u *f(v,v)*a0 *a3*a7 *b2*v - ---*u *f(v,v)*a0 *a3*a7 *b25*v
             2   x                           2   x

             3              3      3             5   2
          + ---*u *f(v,v)*a0 *a3*a7 *b1*v + u *a0 *a7 *b2*u*v
             2   x                           x

                   5   2                  4   3                7
          - 3*u *a0 *a7 *b25*u*v + 3*u *a0 *a7 *b1*u*v - v  *a0 *b25
               x                      x                   3x

                  6          3               5
          + v  *a0 *a7*b1 + ---*v  *f(v,v)*a0 *a3*a7*b25
             3x              2   2x

             3               4      2              6
          - ---*v  *f(v,v)*a0 *a3*a7 *b1 + 3*v  *a0 *a7*b25*u
             2   2x                           2x

                    5   2                       5
          - 3*v  *a0 *a7 *b1*u + 3*v *f(v,v )*a0 *a3*a7*b25
               2x                   x      x

                           4      2       3           2   3   2   2
          - 3*v *f(v,v )*a0 *a3*a7 *b1 - ---*v *f(v,v) *a0 *a3 *a7 *b25
               x      x                   4   x

             3           2   2   2   3                    4      2
          + ---*v *f(v,v) *a0 *a3 *a7 *b1 - 3*v *f(v,v)*a0 *a3*a7 *b25*u
             4   x                             x

                          3      3               5   2      2
          + 3*v *f(v,v)*a0 *a3*a7 *b1*u - 3*v *a0 *a7 *b25*u
               x                             x

                   4   3     2              5
          + 3*v *a0 *a7 *b1*u  + f(v ,v )*a0 *a3*a7*b25*v
               x                    x  x

                       5                1                   3   2   2
          + f(v,v  )*a0 *a3*a7*b25*v + ---*f(v,v )*f(v,v)*a0 *a3 *a7 *b2*v
                 2x                     2       x

             3                   3   2   2
          - ---*f(v,v )*f(v,v)*a0 *a3 *a7 *b25*v
             2       x

             3                   2   2   3                  4      2
          + ---*f(v,v )*f(v,v)*a0 *a3 *a7 *b1*v + f(v,v )*a0 *a3*a7 *b2*u*v
             2       x                                 x

                        4      2                       3      3
          - 3*f(v,v )*a0 *a3*a7 *b25*u*v + 3*f(v,v )*a0 *a3*a7 *b1*u*v
                   x                              x

             1         3   2   3   2         1        3      3   3
          + ----*f(v,v) *a0 *a3 *a7 *b4*v + ---*f(v,v) *a0*a3 *a7 *b25*v
             24                              8

             1        3   3   4         1        2   3   2   2
          - ---*f(v,v) *a3 *a7 *b1*v + ---*f(v,v) *a0 *a3 *a7 *b4*u*v
             8                          4

             3        2   2   2   3            3        2      2   4
          + ---*f(v,v) *a0 *a3 *a7 *b25*u*v - ---*f(v,v) *a0*a3 *a7 *b1*u*v
             4                                 4

             1           4      2     2      3           3      3      2
          + ---*f(v,v)*a0 *a3*a7 *b4*u *v + ---*f(v,v)*a0 *a3*a7 *b25*u *v
             2                               2

             3           2      4     2      1    5   2     3
          - ---*f(v,v)*a0 *a3*a7 *b1*u *v + ---*a0 *a7 *b4*u *v
             2                               3

              4   3      3       3   4     3       6
          + a0 *a7 *b25*u *v - a0 *a7 *b1*u *v)/(a0 *a7)