Solution 5 to problem N1t6s14f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t6s14f1

Expressions

The solution is given through the following expressions:


q54=0


q53=0


q52=0


q51=0


q50=0


      7
     ---*p2*q55
      3
q49=------------
         p7


q48=0


      35
     ----*p2*q55
      3
q47=-------------
         p7


q46=0


q45=0


q44=0


      56
     ----*p2*q55
      3
q43=-------------
         p7


      49
     ----*p2*q55
      3
q42=-------------
         p7


q41=0


      7    2
     ---*p2 *q55
      3
q40=-------------
           2
         p7


q39=0


q38=0


q37=0


q36=0


q35=0


          2
     14*p2 *q55
q34=------------
          2
        p7


     42*p2*q55
q33=-----------
        p7


q32=0


      49
     ----*p2*q55
      3
q31=-------------
         p7


q30=0


q29=0


q28=0


q27=0


q26=0


      28
     ----*p2*q55
      3
q25=-------------
         p7


q24=0


      77
     ----*p2*q55
      3
q23=-------------
         p7


      56
     ----*p2*q55
      3
q22=-------------
         p7


      140    2
     -----*p2 *q55
       9
q21=---------------
            2
          p7


q20=0


q19=0


q18=0


      91    2
     ----*p2 *q55
      3
q17=--------------
           2
         p7


      35    3
     ----*p2 *q55
      27
q16=--------------
           3
         p7


q15=0


      49    2
     ----*p2 *q55
      3
q14=--------------
           2
         p7


q13=0


      35    3
     ----*p2 *q55
      9
q12=--------------
           3
         p7


q11=0


      98    2
     ----*p2 *q55
      3
q10=--------------
           2
         p7


q9=0


q8=0


     35
    ----*p2*q55
     3
q7=-------------
        p7


q6=0


     49    2
    ----*p2 *q55
     9
q5=--------------
          2
        p7


    7*p2*q55
q4=----------
       p7


q3=0


     7
    ---*p2*q55
     3
q2=------------
        p7


q1=0


p6=0


p5=0


p4=p2


p3=0


p1=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:

 q55, p2, 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.

 
{p4,

 g0120*p7 + g0123*p2 + g0125*p2,

 p7,

 p2,

 q55,

            3                 2                  2                  2
 27*g0065*p7  + 63*g0071*p2*p7  + 315*g0073*p2*p7  + 504*g0077*p2*p7

                   2              2                  2                      2
  + 441*g0078*p2*p7  + 63*g0080*p2 *p7 + 378*g0086*p2 *p7 + 1134*g0087*p2*p7

                   2                  2                  2                  2
  + 441*g0089*p2*p7  + 252*g0095*p2*p7  + 693*g0097*p2*p7  + 504*g0098*p2*p7

                2                  2                 3               2
  + 420*g0099*p2 *p7 + 819*g0103*p2 *p7 + 35*g0104*p2  + 441*g0106*p2 *p7

                3               2                     2               2
  + 105*g0108*p2  + 882*g0110*p2 *p7 + 315*g0113*p2*p7  + 147*g0115*p2 *p7

                   2                 2
  + 189*g0116*p2*p7  + 63*g0118*p2*p7 ,

 q17,

            2                   2                   2                   2
 63*g0010*p7 *q55 + 315*g0012*p7 *q55 + 504*g0016*p7 *q55 + 441*g0017*p7 *q55

                                                            2
  + 63*g0019*p2*p7*q55 + 378*g0025*p2*p7*q55 + 1134*g0026*p7 *q55

                2                   2                   2
  + 441*g0028*p7 *q55 + 252*g0034*p7 *q55 + 693*g0036*p7 *q55

                2
  + 504*g0037*p7 *q55 + 420*g0038*p2*p7*q55 + 819*g0042*p2*p7*q55

               2                                         2
  + 35*g0043*p2 *q55 + 441*g0045*p2*p7*q55 + 105*g0047*p2 *q55

                                      2
  + 882*g0049*p2*p7*q55 + 315*g0052*p7 *q55 + 147*g0054*p2*p7*q55

                2                  2                  3              3
  + 189*g0055*p7 *q55 + 63*g0057*p7 *q55 + 27*g0061*p7  + 27*g0063*p7 }

Relevance for the application:

The equation:


                     2
f =Df *Df*f*p2 + (Df) *f *p2 + f  *p7
 t   x                  x       3x

The symmetry:

     7                 2                         2
f =(---*Df  *Df*f*p2*p7 *q55 + 7*Df  *Df *f*p2*p7 *q55
 s   3    5x                       4x   x

        28                  2        35                   2
     + ----*Df  *Df*f *p2*p7 *q55 + ----*Df  *Df  *f*p2*p7 *q55
        3     4x     x               3     3x   2x

        77                   2        49           3     2
     + ----*Df  *Df *f *p2*p7 *q55 + ----*Df  *(Df) *f*p2 *p7*q55
        3     3x   x  x               9     3x

        49                   2        56        2         2
     + ----*Df  *Df*f  *p2*p7 *q55 + ----*(Df  ) *f *p2*p7 *q55
        3     3x     2x               3      2x    x

        98               2     2                               2
     + ----*Df  *Df *(Df) *f*p2 *p7*q55 + 42*Df  *Df *f  *p2*p7 *q55
        3     2x   x                           2x   x  2x

        140           3      2           56                   2
     + -----*Df  *(Df) *f *p2 *p7*q55 + ----*Df  *Df*f  *p2*p7 *q55
         9     2x        x               3     2x     3x

        49       3        2           91       2     2      2
     + ----*(Df ) *Df*f*p2 *p7*q55 + ----*(Df ) *(Df) *f *p2 *p7*q55
        3      x                      3      x          x

        49       2          2        35          5     3
     + ----*(Df ) *f  *p2*p7 *q55 + ----*Df *(Df) *f*p2 *q55
        3      x    3x               9     x

                  3       2           35                  2
     + 14*Df *(Df) *f  *p2 *p7*q55 + ----*Df *Df*f  *p2*p7 *q55
            x        2x               3     x     4x

        35      6      3        7      4       2
     + ----*(Df) *f *p2 *q55 + ---*(Df) *f  *p2 *p7*q55
        27         x            3         3x

        7      2          2             3        3
     + ---*(Df) *f  *p2*p7 *q55 + f  *p7 *q55)/p7
        3         5x               7x

And now in machine readable form:

The system:

df(f(1),t)=d(1,df(f(1),x))*d(1,f(1))*f(1)*p2 + d(1,f(1))**2*df(f(1),x)*p2 + df(f
(1),x,3)*p7$

The symmetry:

df(f(1),s)=(7/3*d(1,df(f(1),x,5))*d(1,f(1))*f(1)*p2*p7**2*q55 + 7*d(1,df(f(1),x,
4))*d(1,df(f(1),x))*f(1)*p2*p7**2*q55 + 28/3*d(1,df(f(1),x,4))*d(1,f(1))*df(f(1)
,x)*p2*p7**2*q55 + 35/3*d(1,df(f(1),x,3))*d(1,df(f(1),x,2))*f(1)*p2*p7**2*q55 + 
77/3*d(1,df(f(1),x,3))*d(1,df(f(1),x))*df(f(1),x)*p2*p7**2*q55 + 49/9*d(1,df(f(1
),x,3))*d(1,f(1))**3*f(1)*p2**2*p7*q55 + 49/3*d(1,df(f(1),x,3))*d(1,f(1))*df(f(1
),x,2)*p2*p7**2*q55 + 56/3*d(1,df(f(1),x,2))**2*df(f(1),x)*p2*p7**2*q55 + 98/3*d
(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1))**2*f(1)*p2**2*p7*q55 + 42*d(1,df(f(1)
,x,2))*d(1,df(f(1),x))*df(f(1),x,2)*p2*p7**2*q55 + 140/9*d(1,df(f(1),x,2))*d(1,f
(1))**3*df(f(1),x)*p2**2*p7*q55 + 56/3*d(1,df(f(1),x,2))*d(1,f(1))*df(f(1),x,3)*
p2*p7**2*q55 + 49/3*d(1,df(f(1),x))**3*d(1,f(1))*f(1)*p2**2*p7*q55 + 91/3*d(1,df
(f(1),x))**2*d(1,f(1))**2*df(f(1),x)*p2**2*p7*q55 + 49/3*d(1,df(f(1),x))**2*df(f
(1),x,3)*p2*p7**2*q55 + 35/9*d(1,df(f(1),x))*d(1,f(1))**5*f(1)*p2**3*q55 + 14*d(
1,df(f(1),x))*d(1,f(1))**3*df(f(1),x,2)*p2**2*p7*q55 + 35/3*d(1,df(f(1),x))*d(1,
f(1))*df(f(1),x,4)*p2*p7**2*q55 + 35/27*d(1,f(1))**6*df(f(1),x)*p2**3*q55 + 7/3*
d(1,f(1))**4*df(f(1),x,3)*p2**2*p7*q55 + 7/3*d(1,f(1))**2*df(f(1),x,5)*p2*p7**2*
q55 + df(f(1),x,7)*p7**3*q55)/p7**3$