Solution 33 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
     ----*a12 *a13*a5 *b4 - ---*a13 *a5 *b24
      24                     8
b42=-----------------------------------------
                         6
                      a12


         1     2       2       3     3   2
      - ---*a12 *a13*a5 *b4 + ---*a13 *a5 *b24
         4                     4
b41=-------------------------------------------
                          5
                       a12


      3     2   2
     ---*a13 *a5 *b24
      2
b40=------------------
              4
           a12


      3     2   2
     ---*a13 *a5 *b24
      4
b39=------------------
              4
           a12


      1     2              3     3
     ---*a12 *a13*a5*b4 - ---*a13 *a5*b24
      2                    2
b38=--------------------------------------
                        4
                     a12


             2
      - 3*a13 *a5*b24
b37=------------------
              3
           a12


             2
      - 3*a13 *a5*b24
b36=------------------
              3
           a12


         3     2
      - ---*a13 *a5*b24
         2
b35=--------------------
               3
            a12


      - a13*a5*b24
b34=---------------
            2
         a12


      - a13*a5*b24
b33=---------------
            2
         a12


      - 3*a13*a5*b24
b32=-----------------
             2
          a12


         3
      - ---*a13*a5*b24
         2
b31=-------------------
              2
           a12


         1     2             3
      - ---*a12 *a13*b4 + a13 *b24
         3
b30=-------------------------------
                    3
                 a12


          2
     3*a13 *b24
b29=------------
           2
        a12


          2
     3*a13 *b24
b28=------------
           2
        a12


     3*a13*b24
b27=-----------
        a12


     3*a13*b24
b26=-----------
        a12


     a13*b24
b25=---------
       a12


      1      2       4       1     3   4
     ----*a12 *a13*a5 *b4 - ---*a13 *a5 *b24
      24                     8
b23=-----------------------------------------
                         7
                      a12


         1     2       3       3     3   3
      - ---*a12 *a13*a5 *b4 + ---*a13 *a5 *b24
         4                     4
b22=-------------------------------------------
                          6
                       a12


         1     2   3       9     2   3
      - ---*a12 *a5 *b4 + ---*a13 *a5 *b24
         4                 4
b21=---------------------------------------
                        5
                     a12


      1     2       2       3     3   2
     ---*a12 *a13*a5 *b4 - ---*a13 *a5 *b24
      2                     2
b20=----------------------------------------
                         5
                      a12


        2   2           2   2
     a12 *a5 *b4 - 6*a13 *a5 *b24
b19=------------------------------
                    4
                 a12


      1     2   2       3     2   2
     ---*a12 *a5 *b4 - ---*a13 *a5 *b24
      4                 2
b18=------------------------------------
                       4
                    a12


                2
      - 3*a13*a5 *b24
b17=------------------
              3
           a12


         5        2
      - ---*a13*a5 *b24
         2
b16=--------------------
               3
            a12


              2
      - a13*a5 *b24
b15=----------------
             3
          a12


         1     2                3
      - ---*a12 *a13*a5*b4 + a13 *a5*b24
         3
b14=-------------------------------------
                       4
                    a12


           2              2
      - a12 *a5*b4 + 3*a13 *a5*b24
b13=-------------------------------
                    3
                 a12


           2              2
      - a12 *a5*b4 + 3*a13 *a5*b24
b12=-------------------------------
                    3
                 a12


     3*a13*a5*b24
b11=--------------
            2
         a12


b10=0


    3*a13*a5*b24
b9=--------------
           2
        a12


    a13*a5*b24
b8=------------
          2
       a12


    a5*b24
b7=--------
     a12


b6=0


b5=0


b3=0


b2=0


b1=0


      1     2   2    1           2
     ---*a13 *a5  - ---*a13*a2*a5
      4              8
a19=-------------------------------
                    3
                 a12


           2       1
      - a13 *a5 + ---*a13*a2*a5
                   2
a18=----------------------------
                   2
                a12


      - a13*a5
a17=-----------
        a12


      - a13*a5
a16=-----------
        a12


        2    1
     a13  - ---*a13*a2
             2
a15=-------------------
            a12


a14=2*a13


      1     2   3    1           3
     ---*a13 *a5  - ---*a13*a2*a5
      4              8
a11=-------------------------------
                    4
                 a12


           2   2    1           2
      - a13 *a5  + ---*a13*a2*a5
                    2
a10=------------------------------
                    3
                 a12


               2    1       2
     - 2*a13*a5  + ---*a2*a5
                    2
a9=---------------------------
                 2
              a12


       2       1
    a13 *a5 - ---*a13*a2*a5
               2
a8=-------------------------
                2
             a12


    2*a13*a5 - a2*a5
a7=------------------
          a12


              1
    a13*a5 - ---*a2*a5
              2
a6=--------------------
           a12


a4=0


a3=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:

 b4,b24,a5,a2,a13,a12

Relevance for the application:

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

                      3           1               3               4
df(u,t)=(u *f(v,v)*a12 *a13*a5 - ---*u *f(v,v)*a12 *a2*a5 + u *a12 *a2*u
          x                       2   x                      x

                        4                          2       2
          + f(v,v  )*a12 *a5 - 2*f(v,v )*f(v,v)*a12 *a13*a5
                 2x                   x

             1                    2      2                3
          + ---*f(v,v )*f(v,v)*a12 *a2*a5  + 2*f(v,v )*a12 *a13*a5*u
             2       x                              x

                       3            1        3    2   3    1        3          3
          - f(v,v )*a12 *a2*a5*u + ---*f(v,v) *a13 *a5  - ---*f(v,v) *a13*a2*a5
                 x                  4                      8

                  2        2   2      1        2              2
          - f(v,v) *a12*a13 *a5 *u + ---*f(v,v) *a12*a13*a2*a5 *u
                                      2

                      2    2     2    1            2            2     4
          + f(v,v)*a12 *a13 *a5*u  - ---*f(v,v)*a12 *a13*a2*a5*u )/a12
                                      2

               3                4                2                  3
df(v,t)=(u *a12 *a13*v + v  *a12  - v *f(v,v)*a12 *a13*a5 + 2*v *a12 *a13*u
          x               2x         x                         x

                       2             1        2    2   2
          - f(v,v )*a12 *a13*a5*v + ---*f(v,v) *a13 *a5 *v
                 x                   4

             1        2          2                   2
          - ---*f(v,v) *a13*a2*a5 *v - f(v,v)*a12*a13 *a5*u*v
             8

             1                                2    2  2      1     2         2
          + ---*f(v,v)*a12*a13*a2*a5*u*v + a12 *a13 *u *v - ---*a12 *a13*a2*u *v
             2                                               2

              3
         )/a12

                       5                              5
df(u,s)=(u  *f(v,v)*a12 *a13*a5*b24 + 3*u *f(v,v )*a12 *a13*a5*b24
          2x                             x      x

             1           2    5   2       3           2    3    2   2
          + ---*u *f(v,v) *a12 *a5 *b4 - ---*u *f(v,v) *a12 *a13 *a5 *b24
             4   x                        2   x

                         6                          4    2
          - u *f(v,v)*a12 *a5*b4*u + 3*u *f(v,v)*a12 *a13 *a5*b24*u
             x                          x

                  7     2                      4       2
          + u *a12 *b4*u  - f(v ,v )*f(v,v)*a12 *a13*a5 *b24
             x                 x  x

                        6           5                     4       2
          + f(v,v  )*a12 *a5*b24 - ---*f(v,v  )*f(v,v)*a12 *a13*a5 *b24
                 3x                 2       2x

                          5                         2    4       2
          + 3*f(v,v  )*a12 *a13*a5*b24*u - 3*f(v,v ) *a12 *a13*a5 *b24
                   2x                             x

             1                2    4   3
          - ---*f(v,v )*f(v,v) *a12 *a5 *b4
             4       x

             9                2    2    2   3                         5   2
          + ---*f(v,v )*f(v,v) *a12 *a13 *a5 *b24 + f(v,v )*f(v,v)*a12 *a5 *b4*u
             4       x                                   x

                                3    2   2                    6        2
          - 6*f(v,v )*f(v,v)*a12 *a13 *a5 *b24*u - f(v,v )*a12 *a5*b4*u
                   x                                    x

                         4    2         2    1         4    2       4
          + 3*f(v,v )*a12 *a13 *a5*b24*u  + ----*f(v,v) *a12 *a13*a5 *b4
                   x                         24

             1        4    3   4        1        3    3       3
          - ---*f(v,v) *a13 *a5 *b24 - ---*f(v,v) *a12 *a13*a5 *b4*u
             8                          4

             3        3        3   3          1        2    4       2     2
          + ---*f(v,v) *a12*a13 *a5 *b24*u + ---*f(v,v) *a12 *a13*a5 *b4*u
             4                                2

             3        2    2    3   2      2    1            5            3
          - ---*f(v,v) *a12 *a13 *a5 *b24*u  - ---*f(v,v)*a12 *a13*a5*b4*u
             2                                  3

                      3    3         3     7
          + f(v,v)*a12 *a13 *a5*b24*u )/a12

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

             3               3    2                    4    2
          - ---*u *f(v,v)*a12 *a13 *a5*b24*v + 3*u *a12 *a13 *b24*u*v
             2   x                                x

                   6        3                4                       5
          + v  *a12 *b24 - ---*v  *f(v,v)*a12 *a13*a5*b24 + 3*v  *a12 *a13*b24*u
             3x             2   2x                             2x

                            4               3           2    2    2   2
          - 3*v *f(v,v )*a12 *a13*a5*b24 + ---*v *f(v,v) *a12 *a13 *a5 *b24
               x      x                     4   x

                           3    2                    4    2      2
          - 3*v *f(v,v)*a12 *a13 *a5*b24*u + 3*v *a12 *a13 *b24*u
               x                                x

                        4                            4
          - f(v ,v )*a12 *a13*a5*b24*v - f(v,v  )*a12 *a13*a5*b24*v
               x  x                           2x

             3                    2    2   2
          + ---*f(v,v )*f(v,v)*a12 *a13 *a5 *b24*v
             2       x

                         3    2               1         3    2       3
          - 3*f(v,v )*a12 *a13 *a5*b24*u*v + ----*f(v,v) *a12 *a13*a5 *b4*v
                   x                          24

             1        3    3   3          1        2    3       2
          - ---*f(v,v) *a13 *a5 *b24*v - ---*f(v,v) *a12 *a13*a5 *b4*u*v
             8                            4

             3        2        3   2            1            4            2
          + ---*f(v,v) *a12*a13 *a5 *b24*u*v + ---*f(v,v)*a12 *a13*a5*b4*u *v
             4                                  2

             3            2    3         2      1     5         3
          - ---*f(v,v)*a12 *a13 *a5*b24*u *v - ---*a12 *a13*b4*u *v
             2                                  3

               3    3      3       6
          + a12 *a13 *b24*u *v)/a12