Solution 1 to problem N1t8s10f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t8s10f1

Expressions

The solution is given through the following expressions: 

       4
     p7 *q8
q21=---------
           4
     405*p6


q20=0


       3
     p7 *q8
q19=--------
          3
     27*p6


         3
     2*p7 *q8
q18=----------
           3
      27*p6


         2
     2*p7 *q8
q17=----------
          2
      9*p6


     2*p7*q8
q16=---------
      3*p6


         2
     2*p7 *q8
q15=----------
          2
      3*p6


         3
     2*p7 *q8
q14=----------
           3
      27*p6


q13=0


       3
     p7 *q8
q12=--------
          3
     27*p6


         2
     4*p7 *q8
q11=----------
          2
      9*p6


       2
     p7 *q8
q10=--------
         2
     3*p6


q7=0


q6=0


q5=0


q4=0


q3=0


q2=0


q1=0


         3
       p7
p13=---------
           2
     108*p6


p12=0


       2
     p7
p11=------
     9*p6


     p7
p10=----
     2


      2
    p7
p9=------
    6*p6


      2
    p7
p8=------
    9*p6


p5=0


p4=0


p3=0


p2=0


p1=0


    2*p7*q8
q9=---------
      p6

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: 
 q8, p6, p7

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. 
 
         4                 3                 3              2   2
{g0037*p7  + 15*g0039*p6*p7  + 30*g0040*p6*p7  + 90*g0041*p6 *p7

                3                  2   2                 3                 3
  + 270*g0042*p6 *p7 + 270*g0043*p6 *p7  + 30*g0044*p6*p7  + 15*g0046*p6*p7

                2   2               2   2               3                  4
  + 180*g0047*p6 *p7  + 135*g0048*p6 *p7  + 810*g0049*p6 *p7 + 405*g0050*p6 ,

 q8,

 p6,

         3                 2              2                    2
 g0058*p7  + 12*g0060*p6*p7  + 54*g0061*p6 *p7 + 18*g0062*p6*p7

                  2               2                  3
  + 12*g0063*p6*p7  + 108*g0064*p6 *p7 + 108*g0065*p6 }

Relevance for the application:



The equation: 



     1               2               2       1               2       3      3
f =(---*Df  *f *p6*p7  + Df *Df*f *p6 *p7 + ---*Df *f  *p6*p7  + (Df) *f *p6
 t   9    2x  x            x     x           6    x  2x                 x

        1      2       2       1              2     1         3    2
     + ---*(Df) *f  *p6 *p7 + ---*Df*f  *p6*p7  + -----*f  *p7 )/p6
        2         2x           9      3x           108   4x


The symmetry: 

     1                3       4               2   2       2                 3
f =(----*Df  *f *p6*p7 *q8 + ---*Df  *Df*f *p6 *p7 *q8 + ----*Df  *f  *p6*p7 *q8
 s   27    3x  x              9    2x     x               27    2x  2x

        1       2      2   2                2      3
     + ---*(Df ) *f *p6 *p7 *q8 + 2*Df *(Df) *f *p6 *p7*q8
        3     x    x                  x        x

        2               2   2       2                3          4      4
     + ---*Df *Df*f  *p6 *p7 *q8 + ----*Df *f  *p6*p7 *q8 + (Df) *f *p6 *q8
        3    x     2x               27    x  3x                    x

        2      3       3          2      2       2   2
     + ---*(Df) *f  *p6 *p7*q8 + ---*(Df) *f  *p6 *p7 *q8
        3         2x              9         3x

        1               3        1         4       4
     + ----*Df*f  *p6*p7 *q8 + -----*f  *p7 *q8)/p6
        27      4x              405   5x


And now in machine readable form: 

The system: 

df(f(1),t)=(1/9*d(1,df(f(1),x,2))*df(f(1),x)*p6*p7**2 + d(1,df(f(1),x))*d(1,f(1)
)*df(f(1),x)*p6**2*p7 + 1/6*d(1,df(f(1),x))*df(f(1),x,2)*p6*p7**2 + d(1,f(1))**3
*df(f(1),x)*p6**3 + 1/2*d(1,f(1))**2*df(f(1),x,2)*p6**2*p7 + 1/9*d(1,f(1))*df(f(
1),x,3)*p6*p7**2 + 1/108*df(f(1),x,4)*p7**3)/p6**2$


The symmetry: 

df(f(1),s)=(1/27*d(1,df(f(1),x,3))*df(f(1),x)*p6*p7**3*q8 + 4/9*d(1,df(f(1),x,2)
)*d(1,f(1))*df(f(1),x)*p6**2*p7**2*q8 + 2/27*d(1,df(f(1),x,2))*df(f(1),x,2)*p6*
p7**3*q8 + 1/3*d(1,df(f(1),x))**2*df(f(1),x)*p6**2*p7**2*q8 + 2*d(1,df(f(1),x))*
d(1,f(1))**2*df(f(1),x)*p6**3*p7*q8 + 2/3*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)
*p6**2*p7**2*q8 + 2/27*d(1,df(f(1),x))*df(f(1),x,3)*p6*p7**3*q8 + d(1,f(1))**4*
df(f(1),x)*p6**4*q8 + 2/3*d(1,f(1))**3*df(f(1),x,2)*p6**3*p7*q8 + 2/9*d(1,f(1))
**2*df(f(1),x,3)*p6**2*p7**2*q8 + 1/27*d(1,f(1))*df(f(1),x,4)*p6*p7**3*q8 + 1/
405*df(f(1),x,5)*p7**4*q8)/p6**4$