Solution 2 to problem N2t6s8f3

Remaining equations | Expressions | Parameters | Inequalities | Relevance | Back to problem N2t6s8f3

Equations

The following unsolved equations remain: 
    2     2
0=p1  + p2

Expressions

The solution is given through the following expressions: 

q15=0


q14=0


      3     6       25     4   2       5     2   4        1     6
     ----*p1 *q5 - -----*p1 *p2 *q5 - ----*p1 *p2 *q5 - -----*p2 *q5
      16            192                72                576
q13=-----------------------------------------------------------------
                                   4   2
                                 p1 *p2


         1    4       37     2   2        1     4
      - ---*p1 *q5 + -----*p1 *p2 *q5 + -----*p2 *q5
         8            288                288
q12=-------------------------------------------------
                           3
                         p1 *p2


      3     6       25     4   2       5     2   4        1     6
     ----*p1 *q5 - -----*p1 *p2 *q5 - ----*p1 *p2 *q5 - -----*p2 *q5
      16            192                72                576
q11=-----------------------------------------------------------------
                                   5
                                 p1 *p2


         1    4       37     2   2        1     4
      - ---*p1 *q5 + -----*p1 *p2 *q5 + -----*p2 *q5
         8            288                288
q10=-------------------------------------------------
                             4
                           p1


     9    6       23    4   2       7     2   4        1     6
    ---*p1 *q5 + ----*p1 *p2 *q5 + ----*p1 *p2 *q5 + -----*p2 *q5
     8            32                72                288
q9=---------------------------------------------------------------
                                 4   2
                               p1 *p2


     27    10       123    8   2       1433    6   4       1481    4   6
q8=(----*p1  *q5 + -----*p1 *p2 *q5 + ------*p1 *p2 *q5 + ------*p1 *p2 *q5
     8              16                 384                 3456

         31     2   8        1      10        7   3    37    5   5
     + ------*p1 *p2 *q5 - ------*p2  *q5)/(p1 *p2  - ----*p1 *p2
        3456                3456                       36

       1     3   7
    - ----*p1 *p2 )
       36


q7=0


q6=0


    p2*q5
q4=-------
     p1


q3=0


q2=0


q1=0


p8=0


p7=0


     7     6    223    4   2    11     2   4     1     6
    ----*p1  - -----*p1 *p2  - -----*p1 *p2  - -----*p2
     16         576             144             576
p6=------------------------------------------------------
                             4
                           p1 *p2


     3    4    37    2   2    1     4
    ---*p1  - ----*p1 *p2  - ----*p2
     8         96             96
p5=-----------------------------------
                      2
                 p1*p2


     3     6    25     4   2    5     2   4     1     6
    ----*p1  - -----*p1 *p2  - ----*p1 *p2  - -----*p2
     16         192             72             576
p4=-----------------------------------------------------
                            3   2
                          p1 *p2


        1    4    37     2   2     1     4
     - ---*p1  + -----*p1 *p2  + -----*p2
        8         288             288
p3=----------------------------------------
                      2
                    p1 *p2

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: 
 q5, p2, p1

Inequalities

In the following not identically vanishing expressions are shown. Any auxiliary variables g00?? are used to express that at least one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3 means that either p4 or p3 or both are non-vanishing. 
 
{p2,

 p1,

 q5,

      4        2   2     4
 36*p1  - 37*p1 *p2  - p2 ,

 p3,

             6                  4   3              2   5           7
 252*g0069*p1 *p2 - 223*g0069*p1 *p2  - 44*g0069*p1 *p2  - g0069*p2

                7               5   2             3   4               7
  + 216*g0070*p1  - 222*g0070*p1 *p2  - 6*g0070*p1 *p2  + 108*g0071*p1

               5   2              3   4              6              6
  - 75*g0071*p1 *p2  - 40*g0071*p1 *p2  - g0071*p1*p2  - 72*g0072*p1 *p2

               4   3             2   5               4   3               5   2
  + 74*g0072*p1 *p2  + 2*g0072*p1 *p2  + 576*g0073*p1 *p2  + 576*g0074*p1 *p2 ,

             6                 4   2                 2   4              6
 108*g0004*p1 *q5 - 75*g0004*p1 *p2 *q5 - 40*g0004*p1 *p2 *q5 - g0004*p2 *q5

               5                    3   3                   5
  - 72*g0005*p1 *p2*q5 + 74*g0005*p1 *p2 *q5 + 2*g0005*p1*p2 *q5

               4   2                 2   4                6
  - 72*g0006*p1 *p2 *q5 + 74*g0006*p1 *p2 *q5 + 2*g0006*p2 *q5

                6                  4   2                 2   4
  + 648*g0007*p1 *q5 + 414*g0007*p1 *p2 *q5 + 56*g0007*p1 *p2 *q5

              6                  4   2                  7               5   2
  + 2*g0007*p2 *q5 + 576*g0010*p1 *p2 *q5 + 216*g0014*p1  - 222*g0014*p1 *p2

              3   4               7              5   2              3   4
  - 6*g0014*p1 *p2  + 108*g0015*p1  - 75*g0015*p1 *p2  - 40*g0015*p1 *p2

               6              6                 4   3             2   5
  - g0015*p1*p2  - 72*g0016*p1 *p2 + 74*g0016*p1 *p2  + 2*g0016*p1 *p2

                5   2
  + 576*g0017*p1 *p2 ,

              11                      9   3                   7   5
 3888*g0019*p1  *p2*q5 - 6696*g0019*p1 *p2 *q5 + 1227*g0019*p1 *p2 *q5

                 5   7                 3   9                 11
  + 1519*g0019*p1 *p2 *q5 + 77*g0019*p1 *p2 *q5 + g0019*p1*p2  *q5

                 10   2                   8   4                   6   6
  - 2592*g0020*p1  *p2 *q5 + 5328*g0020*p1 *p2 *q5 - 2594*g0020*p1 *p2 *q5

                4   8                2   10                   10   2
  - 148*g0020*p1 *p2 *q5 - 2*g0020*p1 *p2  *q5 + 3888*g0021*p1  *p2 *q5

                 8   4                   6   6                   4   8
  - 6696*g0021*p1 *p2 *q5 + 1227*g0021*p1 *p2 *q5 + 1519*g0021*p1 *p2 *q5

               2   10              12                    12
  + 77*g0021*p1 *p2  *q5 + g0021*p2  *q5 + 69984*g0022*p1  *q5

                   10   2                    8   4                   6   6
  + 159408*g0022*p1  *p2 *q5 + 77382*g0022*p1 *p2 *q5 + 8886*g0022*p1 *p2 *q5

                4   8                2   10                    8   4
  + 186*g0022*p1 *p2 *q5 - 6*g0022*p1 *p2  *q5 + 20736*g0025*p1 *p2 *q5

                  6   6                  4   8                   11   2
  - 21312*g0025*p1 *p2 *q5 - 576*g0025*p1 *p2 *q5 + 9072*g0029*p1  *p2

                  9   4                7   6                5   8
  - 17352*g0029*p1 *p2  + 6415*g0029*p1 *p2  + 1815*g0029*p1 *p2

               3   10              12                12                   10   3
  + 81*g0029*p1 *p2   + g0029*p1*p2   + 3888*g0030*p1  *p2 - 6696*g0030*p1  *p2

                 8   5                6   7              4   9           2   11
  + 1227*g0030*p1 *p2  + 1519*g0030*p1 *p2  + 77*g0030*p1 *p2  + g0030*p1 *p2

                 11   2                9   4                7   6
  - 2592*g0031*p1  *p2  + 5328*g0031*p1 *p2  - 2594*g0031*p1 *p2

                5   8             3   10                 9   4
  - 148*g0031*p1 *p2  - 2*g0031*p1 *p2   + 20736*g0032*p1 *p2

                  7   6               5   8
  - 21312*g0032*p1 *p2  - 576*g0032*p1 *p2 ,

              11                      9   3                   7   5
 3888*g0033*p1  *p2*q5 - 6696*g0033*p1 *p2 *q5 + 1227*g0033*p1 *p2 *q5

                 5   7                 3   9                 11
  + 1519*g0033*p1 *p2 *q5 + 77*g0033*p1 *p2 *q5 + g0033*p1*p2  *q5

                 10   2                   8   4                   6   6
  - 2592*g0034*p1  *p2 *q5 + 5328*g0034*p1 *p2 *q5 - 2594*g0034*p1 *p2 *q5

                4   8                2   10                   10   2
  - 148*g0034*p1 *p2 *q5 - 2*g0034*p1 *p2  *q5 + 3888*g0035*p1  *p2 *q5

                 8   4                   6   6                   4   8
  - 6696*g0035*p1 *p2 *q5 + 1227*g0035*p1 *p2 *q5 + 1519*g0035*p1 *p2 *q5

               2   10              12                   9   3
  + 77*g0035*p1 *p2  *q5 + g0035*p2  *q5 - 2592*g0036*p1 *p2 *q5

                 7   5                   5   7                  3   9
  + 5328*g0036*p1 *p2 *q5 - 2594*g0036*p1 *p2 *q5 - 148*g0036*p1 *p2 *q5

                 11                    11                      9   3
  - 2*g0036*p1*p2  *q5 + 23328*g0037*p1  *p2*q5 - 9072*g0037*p1 *p2 *q5

                  7   5                   5   7                  3   9
  - 13950*g0037*p1 *p2 *q5 - 2414*g0037*p1 *p2 *q5 - 130*g0037*p1 *p2 *q5

                 11                    12                     10   2
  - 2*g0037*p1*p2  *q5 + 69984*g0038*p1  *q5 + 159408*g0038*p1  *p2 *q5

                  8   4                   6   6                  4   8
  + 77382*g0038*p1 *p2 *q5 + 8886*g0038*p1 *p2 *q5 + 186*g0038*p1 *p2 *q5

              2   10                    9   3                    7   5
  - 6*g0038*p1 *p2  *q5 + 20736*g0041*p1 *p2 *q5 - 21312*g0041*p1 *p2 *q5

                5   7                    8   4                    6   6
  - 576*g0041*p1 *p2 *q5 + 20736*g0042*p1 *p2 *q5 - 21312*g0042*p1 *p2 *q5

                4   8                   11   2                 9   4
  - 576*g0042*p1 *p2 *q5 + 9072*g0046*p1  *p2  - 17352*g0046*p1 *p2

                 7   6                5   8              3   10              12
  + 6415*g0046*p1 *p2  + 1815*g0046*p1 *p2  + 81*g0046*p1 *p2   + g0046*p1*p2

                 12                    10   3                8   5
  + 7776*g0047*p1  *p2 - 15984*g0047*p1  *p2  + 7782*g0047*p1 *p2

                6   7             4   9                12
  + 444*g0047*p1 *p2  + 6*g0047*p1 *p2  + 3888*g0048*p1  *p2

                 10   3                8   5                6   7
  - 6696*g0048*p1  *p2  + 1227*g0048*p1 *p2  + 1519*g0048*p1 *p2

               4   9           2   11                11   2                9   4
  + 77*g0048*p1 *p2  + g0048*p1 *p2   - 2592*g0049*p1  *p2  + 5328*g0049*p1 *p2

                 7   6               5   8             3   10
  - 2594*g0049*p1 *p2  - 148*g0049*p1 *p2  - 2*g0049*p1 *p2

                  9   4                 7   6               5   8
  + 20736*g0050*p1 *p2  - 21312*g0050*p1 *p2  - 576*g0050*p1 *p2

                  10   3                 8   5               6   7
  + 20736*g0051*p1  *p2  - 21312*g0051*p1 *p2  - 576*g0051*p1 *p2 ,

              11                   9   3                7   5
 3888*g0054*p1  *p2 - 6696*g0054*p1 *p2  + 1227*g0054*p1 *p2

                 5   7              3   9              11                10   2
  + 1519*g0054*p1 *p2  + 77*g0054*p1 *p2  + g0054*p1*p2   - 2592*g0055*p1  *p2

                 8   4                6   6               4   8
  + 5328*g0055*p1 *p2  - 2594*g0055*p1 *p2  - 148*g0055*p1 *p2

              2   10                10   2                8   4
  - 2*g0055*p1 *p2   + 3888*g0056*p1  *p2  - 6696*g0056*p1 *p2

                 6   6                4   8              2   10           12
  + 1227*g0056*p1 *p2  + 1519*g0056*p1 *p2  + 77*g0056*p1 *p2   + g0056*p2

                 9   3                7   5                5   7
  - 2592*g0057*p1 *p2  + 5328*g0057*p1 *p2  - 2594*g0057*p1 *p2

                3   9                11                 11
  - 148*g0057*p1 *p2  - 2*g0057*p1*p2   + 23328*g0058*p1  *p2

                 9   3                 7   5                5   7
  - 9072*g0058*p1 *p2  - 13950*g0058*p1 *p2  - 2414*g0058*p1 *p2

                3   9                11                 12
  - 130*g0058*p1 *p2  - 2*g0058*p1*p2   + 69984*g0059*p1

                   10   2                 8   4                6   6
  + 159408*g0059*p1  *p2  + 77382*g0059*p1 *p2  + 8886*g0059*p1 *p2

                4   8             2   10                 9   3
  + 186*g0059*p1 *p2  - 6*g0059*p1 *p2   + 20736*g0062*p1 *p2

                  7   5               5   7                 8   4
  - 21312*g0062*p1 *p2  - 576*g0062*p1 *p2  + 20736*g0063*p1 *p2

                  6   6               4   8
  - 21312*g0063*p1 *p2  - 576*g0063*p1 *p2 }

Relevance for the application:



The equation: 



             5   2    1              6       37               4   3
f =(D f *f*p1 *p2  - ---*D f*D D f*p1 *p2 + -----*D f*D D f*p1 *p2
 t   2 x              8   2   1 2            288   2   1 2

         1               2   5    3           7    37           5   2
     + -----*D f*D D f*p1 *p2  + ---*D f*f *p1  - ----*D f*f *p1 *p2
        288   2   1 2             8   2   x        96   2   x

        1            3   4    3               7    25               5   2
     - ----*D f*f *p1 *p2  + ----*D D f*D f*p1  - -----*D D f*D f*p1 *p2
        96   2   x            16   1 2   1         192   1 2   1

        5               3   4     1                  6            4   3
     - ----*D D f*D f*p1 *p2  - -----*D D f*D f*p1*p2  + D f *f*p1 *p2
        72   1 2   1             576   1 2   1            1 x

        7            6       223           4   3    11            2   5
     + ----*D f*f *p1 *p2 - -----*D f*f *p1 *p2  - -----*D f*f *p1 *p2
        16   1   x           576   1   x            144   1   x

         1            7     4   2
     - -----*D f*f *p2 )/(p1 *p2 )
        576   1   x


The symmetry: 

              9   3       37            7   5       1             5   7
f =(D f  *f*p1 *p2 *q5 - ----*D f  *f*p1 *p2 *q5 - ----*D f  *f*p1 *p2 *q5
 s   2 2x                 36   2 2x                 36   2 2x

        9            11          7             9   3
     + ---*D f *f *p1  *p2*q5 - ----*D f *f *p1 *p2 *q5
        8   2 x  x               16   2 x  x

        775             7   5       1207             5   7
     - ------*D f *f *p1 *p2 *q5 - -------*D f *f *p1 *p2 *q5
        1152   2 x  x               10368   2 x  x

         65              3   9         1                 11
     - -------*D f *f *p1 *p2 *q5 - -------*D f *f *p1*p2  *q5
        10368   2 x  x               10368   2 x  x

        1               10   2       37                8   4
     - ---*D f*D D f *p1  *p2 *q5 + -----*D f*D D f *p1 *p2 *q5
        8   2   1 2 x                144   2   1 2 x

        1297                6   6        37                4   8
     - -------*D f*D D f *p1 *p2 *q5 - ------*D f*D D f *p1 *p2 *q5
        10368   2   1 2 x               5184   2   1 2 x

          1                 2   10       1            9   3
     - -------*D f*D D f *p1 *p2  *q5 - ---*D f*f  *p1 *p2 *q5
        10368   2   1 2 x                8   2   2x

        37             7   5       1297             5   7
     + -----*D f*f  *p1 *p2 *q5 - -------*D f*f  *p1 *p2 *q5
        144   2   2x               10368   2   2x

         37             3   9         1                 11
     - ------*D f*f  *p1 *p2 *q5 - -------*D f*f  *p1*p2  *q5
        5184   2   2x               10368   2   2x

        3                11          31               9   3
     + ----*D D f *D f*p1  *p2*q5 - ----*D D f *D f*p1 *p2 *q5
        16   1 2 x  1                96   1 2 x  1

        409                7   5       1519                5   7
     + ------*D D f *D f*p1 *p2 *q5 + -------*D D f *D f*p1 *p2 *q5
        6912   1 2 x  1                20736   1 2 x  1

         77                 3   9         1                    11
     + -------*D D f *D f*p1 *p2 *q5 + -------*D D f *D f*p1*p2  *q5
        20736   1 2 x  1                20736   1 2 x  1

                 8   4       37            6   6       1             4   8
     + D f  *f*p1 *p2 *q5 - ----*D f  *f*p1 *p2 *q5 - ----*D f  *f*p1 *p2 *q5
        1 2x                 36   1 2x                 36   1 2x

        27            12       123            10   2
     + ----*D f *f *p1  *q5 + -----*D f *f *p1  *p2 *q5
        8    1 x  x            16    1 x  x

        1433            8   4       1481            6   6
     + ------*D f *f *p1 *p2 *q5 + ------*D f *f *p1 *p2 *q5
        384    1 x  x               3456   1 x  x

         31             4   8        1              2   10
     + ------*D f *f *p1 *p2 *q5 - ------*D f *f *p1 *p2  *q5
        3456   1 x  x               3456   1 x  x

        3             10   2       31            8   4
     + ----*D f*f  *p1  *p2 *q5 - ----*D f*f  *p1 *p2 *q5
        16   1   2x                96   1   2x

        409             6   6       1519             4   8
     + ------*D f*f  *p1 *p2 *q5 + -------*D f*f  *p1 *p2 *q5
        6912   1   2x               20736   1   2x

         77              2   10         1              12        9   3
     + -------*D f*f  *p1 *p2  *q5 + -------*D f*f  *p2  *q5)/(p1 *p2
        20736   1   2x                20736   1   2x

       37    7   5    1     5   7
    - ----*p1 *p2  - ----*p1 *p2 )
       36             36


And now in machine readable form: 

The system: 

df(f(1),t)=(d(2,df(f(1),x))*f(1)*p1**5*p2**2 - 1/8*d(2,f(1))*d(1,d(2,f(1)))*p1**
6*p2 + 37/288*d(2,f(1))*d(1,d(2,f(1)))*p1**4*p2**3 + 1/288*d(2,f(1))*d(1,d(2,f(1
)))*p1**2*p2**5 + 3/8*d(2,f(1))*df(f(1),x)*p1**7 - 37/96*d(2,f(1))*df(f(1),x)*p1
**5*p2**2 - 1/96*d(2,f(1))*df(f(1),x)*p1**3*p2**4 + 3/16*d(1,d(2,f(1)))*d(1,f(1)
)*p1**7 - 25/192*d(1,d(2,f(1)))*d(1,f(1))*p1**5*p2**2 - 5/72*d(1,d(2,f(1)))*d(1,
f(1))*p1**3*p2**4 - 1/576*d(1,d(2,f(1)))*d(1,f(1))*p1*p2**6 + d(1,df(f(1),x))*f(
1)*p1**4*p2**3 + 7/16*d(1,f(1))*df(f(1),x)*p1**6*p2 - 223/576*d(1,f(1))*df(f(1),
x)*p1**4*p2**3 - 11/144*d(1,f(1))*df(f(1),x)*p1**2*p2**5 - 1/576*d(1,f(1))*df(f(
1),x)*p2**7)/(p1**4*p2**2)$


The symmetry: 

df(f(1),s)=(d(2,df(f(1),x,2))*f(1)*p1**9*p2**3*q5 - 37/36*d(2,df(f(1),x,2))*f(1)
*p1**7*p2**5*q5 - 1/36*d(2,df(f(1),x,2))*f(1)*p1**5*p2**7*q5 + 9/8*d(2,df(f(1),x
))*df(f(1),x)*p1**11*p2*q5 - 7/16*d(2,df(f(1),x))*df(f(1),x)*p1**9*p2**3*q5 - 
775/1152*d(2,df(f(1),x))*df(f(1),x)*p1**7*p2**5*q5 - 1207/10368*d(2,df(f(1),x))*
df(f(1),x)*p1**5*p2**7*q5 - 65/10368*d(2,df(f(1),x))*df(f(1),x)*p1**3*p2**9*q5 -
 1/10368*d(2,df(f(1),x))*df(f(1),x)*p1*p2**11*q5 - 1/8*d(2,f(1))*d(1,d(2,df(f(1)
,x)))*p1**10*p2**2*q5 + 37/144*d(2,f(1))*d(1,d(2,df(f(1),x)))*p1**8*p2**4*q5 - 
1297/10368*d(2,f(1))*d(1,d(2,df(f(1),x)))*p1**6*p2**6*q5 - 37/5184*d(2,f(1))*d(1
,d(2,df(f(1),x)))*p1**4*p2**8*q5 - 1/10368*d(2,f(1))*d(1,d(2,df(f(1),x)))*p1**2*
p2**10*q5 - 1/8*d(2,f(1))*df(f(1),x,2)*p1**9*p2**3*q5 + 37/144*d(2,f(1))*df(f(1)
,x,2)*p1**7*p2**5*q5 - 1297/10368*d(2,f(1))*df(f(1),x,2)*p1**5*p2**7*q5 - 37/
5184*d(2,f(1))*df(f(1),x,2)*p1**3*p2**9*q5 - 1/10368*d(2,f(1))*df(f(1),x,2)*p1*
p2**11*q5 + 3/16*d(1,d(2,df(f(1),x)))*d(1,f(1))*p1**11*p2*q5 - 31/96*d(1,d(2,df(
f(1),x)))*d(1,f(1))*p1**9*p2**3*q5 + 409/6912*d(1,d(2,df(f(1),x)))*d(1,f(1))*p1
**7*p2**5*q5 + 1519/20736*d(1,d(2,df(f(1),x)))*d(1,f(1))*p1**5*p2**7*q5 + 77/
20736*d(1,d(2,df(f(1),x)))*d(1,f(1))*p1**3*p2**9*q5 + 1/20736*d(1,d(2,df(f(1),x)
))*d(1,f(1))*p1*p2**11*q5 + d(1,df(f(1),x,2))*f(1)*p1**8*p2**4*q5 - 37/36*d(1,df
(f(1),x,2))*f(1)*p1**6*p2**6*q5 - 1/36*d(1,df(f(1),x,2))*f(1)*p1**4*p2**8*q5 + 
27/8*d(1,df(f(1),x))*df(f(1),x)*p1**12*q5 + 123/16*d(1,df(f(1),x))*df(f(1),x)*p1
**10*p2**2*q5 + 1433/384*d(1,df(f(1),x))*df(f(1),x)*p1**8*p2**4*q5 + 1481/3456*d
(1,df(f(1),x))*df(f(1),x)*p1**6*p2**6*q5 + 31/3456*d(1,df(f(1),x))*df(f(1),x)*p1
**4*p2**8*q5 - 1/3456*d(1,df(f(1),x))*df(f(1),x)*p1**2*p2**10*q5 + 3/16*d(1,f(1)
)*df(f(1),x,2)*p1**10*p2**2*q5 - 31/96*d(1,f(1))*df(f(1),x,2)*p1**8*p2**4*q5 + 
409/6912*d(1,f(1))*df(f(1),x,2)*p1**6*p2**6*q5 + 1519/20736*d(1,f(1))*df(f(1),x,
2)*p1**4*p2**8*q5 + 77/20736*d(1,f(1))*df(f(1),x,2)*p1**2*p2**10*q5 + 1/20736*d(
1,f(1))*df(f(1),x,2)*p2**12*q5)/(p1**9*p2**3 - 37/36*p1**7*p2**5 - 1/36*p1**5*p2
**7)$