Solution 1 to problem N2t6s8f3

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

Equations

The following unsolved equations remain: 
    2       2
0=p3  + 9*p5

Expressions

The solution is given through the following expressions: 

q15=0


q14=0


      1    2
     ---*p3 *q5
      6
q13=------------
          2
        p5


         1
      - ---*p3*q5
         2
q12=--------------
          p5


         1    2       3    2
      - ---*p3 *q5 - ---*p5 *q5
         4            4
q11=----------------------------
               p3*p5


         1     2       1    2
      - ----*p3 *q5 - ---*p5 *q5
         12            4
q10=-----------------------------
                   2
                 p5


        1    2
     - ---*p3 *q5
        6
q9=---------------
           2
         p5


        1
     - ---*p3*q5
        2
q8=--------------
         p5


        1
     - ---*p3*q5
        2
q7=--------------
         p5


     1    2
    ---*p3 *q5
     6
q6=------------
         2
       p5


    3*p5*q5
q4=---------
      p3


q3=0


q2=0


q1=0


p8=0


p7=0


        2
    3*p5
p6=-------
     p3


        1    2
     - ---*p3
        3
p4=------------
        p5


           2
     - 6*p5
p2=----------
       p3


p1= - 2*p5

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, p5, p3

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. 
 
{p1,

 q5,

 p3,

 p5,

 p2,

 q8,

           3                2           3             2                 3
 9*g0069*p5  + 3*g0070*p3*p5  - g0071*p3  + 3*g0072*p3 *p5 - 18*g0073*p5

                 2
  - 6*g0074*p3*p5 ,

 p4,

   2       2
 p3  + 3*p5 ,

           2                                 2                2
 2*g0004*p3 *q5 - 6*g0005*p3*p5*q5 - g0006*p3 *q5 - 3*g0006*p5 *q5

              2                                   2                 2
  - 2*g0007*p3 *q5 - 6*g0008*p3*p5*q5 + 2*g0009*p3 *q5 + 12*g0010*p5 *q5

               3             2                    2              3
  + 12*g0014*p5  - 4*g0015*p3 *p5 + 12*g0016*p3*p5  - 24*g0017*p5 ,

           3                2                   2                   3
 2*g0019*p3 *q5 - 6*g0020*p3 *p5*q5 - 3*g0021*p3 *p5*q5 - 9*g0021*p5 *q5

              2                   2                   3                 3
  - 6*g0022*p3 *p5*q5 - 6*g0023*p3 *p5*q5 + 2*g0024*p3 *q5 + 36*g0025*p5 *q5

               4             3                 2   2              4
  + 36*g0029*p5  - 4*g0030*p3 *p5 + 12*g0031*p3 *p5  - 72*g0032*p5 ,

           3                2                   2                   3
 2*g0033*p3 *q5 - 6*g0034*p3 *p5*q5 - 3*g0035*p3 *p5*q5 - 9*g0035*p5 *q5

            3                   2                3                2
  - g0036*p3 *q5 - 3*g0036*p3*p5 *q5 - 2*g0037*p3 *q5 - 6*g0038*p3 *p5*q5

              2                   3                    2                 3
  - 6*g0039*p3 *p5*q5 + 2*g0040*p3 *q5 + 12*g0041*p3*p5 *q5 + 36*g0042*p5 *q5

               4                 3             3                 2   2
  + 36*g0046*p5  + 12*g0047*p3*p5  - 4*g0048*p3 *p5 + 12*g0049*p3 *p5

               4                 3
  - 72*g0050*p5  - 24*g0051*p3*p5 ,

           3             2                2                3           3
 2*g0054*p3  - 6*g0055*p3 *p5 - 3*g0056*p3 *p5 - 9*g0056*p5  - g0057*p3

                 2             3             2                2                3
  - 3*g0057*p3*p5  - 2*g0058*p3  - 6*g0059*p3 *p5 - 6*g0060*p3 *p5 + 2*g0061*p3

                  2              3
  + 12*g0062*p3*p5  + 36*g0063*p5 }

Relevance for the application:



The equation: 



                     2               2                  2    1              3
f =( - 2*D f *f*p3*p5  + D f*D D f*p3 *p5 + D f*f *p3*p5  - ---*D D f*D f*p3
 t        2 x             2   1 2            2   x           3   1 2   1

                  3              3
     - 6*D f *f*p5  + 3*D f*f *p5 )/(p3*p5)
          1 x            1   x


The symmetry: 

                 2       1               2          1            3
f =(D f  *f*p3*p5 *q5 - ---*D f *D D f*p3 *p5*q5 - ---*D f *f *p3 *q5
 s   2 2x                2   2 x  1 2               6   2 x  x

        1               2          1             3       1               2
     - ---*D f*D D f *p3 *p5*q5 - ----*D f*f  *p3 *q5 - ---*D f*f  *p3*p5 *q5
        2   2   1 2 x              12   2   2x           4   2   2x

        1               3       1               3                  3
     + ---*D D f *D f*p3 *q5 + ---*D D f*D f *p3 *q5 + 3*D f  *f*p5 *q5
        6   1 2 x  1            6   1 2   1 x             1 2x

        1            2          1            2          3            3
     - ---*D f *f *p3 *p5*q5 - ---*D f*f  *p3 *p5*q5 - ---*D f*f  *p5 *q5)/(p3
        2   1 x  x              4   1   2x              4   1   2x

      2
   *p5 )


And now in machine readable form: 

The system: 

df(f(1),t)=( - 2*d(2,df(f(1),x))*f(1)*p3*p5**2 + d(2,f(1))*d(1,d(2,f(1)))*p3**2*
p5 + d(2,f(1))*df(f(1),x)*p3*p5**2 - 1/3*d(1,d(2,f(1)))*d(1,f(1))*p3**3 - 6*d(1,
df(f(1),x))*f(1)*p5**3 + 3*d(1,f(1))*df(f(1),x)*p5**3)/(p3*p5)$


The symmetry: 

df(f(1),s)=(d(2,df(f(1),x,2))*f(1)*p3*p5**2*q5 - 1/2*d(2,df(f(1),x))*d(1,d(2,f(1
)))*p3**2*p5*q5 - 1/6*d(2,df(f(1),x))*df(f(1),x)*p3**3*q5 - 1/2*d(2,f(1))*d(1,d(
2,df(f(1),x)))*p3**2*p5*q5 - 1/12*d(2,f(1))*df(f(1),x,2)*p3**3*q5 - 1/4*d(2,f(1)
)*df(f(1),x,2)*p3*p5**2*q5 + 1/6*d(1,d(2,df(f(1),x)))*d(1,f(1))*p3**3*q5 + 1/6*d
(1,d(2,f(1)))*d(1,df(f(1),x))*p3**3*q5 + 3*d(1,df(f(1),x,2))*f(1)*p5**3*q5 - 1/2
*d(1,df(f(1),x))*df(f(1),x)*p3**2*p5*q5 - 1/4*d(1,f(1))*df(f(1),x,2)*p3**2*p5*q5
 - 3/4*d(1,f(1))*df(f(1),x,2)*p5**3*q5)/(p3*p5**2)$