Solution 3 to problem N1t4s12f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t4s12f1

Expressions

The solution is given through the following expressions:


q34=0


q33=0


     3*p3*q35
q32=----------
        p4


      15    2
     ----*p3 *q35
      4
q31=--------------
           2
         p4


      15
     ----*p3*q35
      2
q30=-------------
         p4


     10*p3*q35
q29=-----------
        p4


          2
     15*p3 *q35
q28=------------
          2
        p4


      5    3
     ---*p3 *q35
      2
q27=-------------
           3
         p4


q26=0


      15    4
     ----*p3 *q35
      16
q25=--------------
           4
         p4


      45    3
     ----*p3 *q35
      4
q24=--------------
           3
         p4


      45    2
     ----*p3 *q35
      4
q23=--------------
           2
         p4


          2
     15*p3 *q35
q22=------------
          2
        p4


      15
     ----*p3*q35
      2
q21=-------------
         p4


q20=0


q19=0


     3*p3*q35
q18=----------
        p4


      15    2
     ----*p3 *q35
      2
q17=--------------
           2
         p4


      15    3
     ----*p3 *q35
      2
q16=--------------
           3
         p4


          2
     15*p3 *q35
q15=------------
          2
        p4


      15    4
     ----*p3 *q35
      4
q14=--------------
           4
         p4


      45    3
     ----*p3 *q35
      4
q13=--------------
           3
         p4


      3     5
     ----*p3 *q35
      16
q12=--------------
           5
         p4


q11=0


q10=0


q9=0


q8=0


q7=0


q6=0


q5=0


q4=0


q3=0


q2=0


q1=0


p2=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:

 q35, p3, p4

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.

 
{g0077*p4 + g0078*p3,

 p4,

 q35,

            5                 4              2   3                  4
 16*g0042*p4  + 48*g0045*p3*p4  + 60*g0046*p3 *p4  + 120*g0047*p3*p4

                   4               2   3              3   2              4
  + 160*g0048*p3*p4  + 240*g0049*p3 *p4  + 40*g0050*p3 *p4  + 15*g0052*p3 *p4

                3   2               2   3               2   3                  4
  + 180*g0053*p3 *p4  + 180*g0054*p3 *p4  + 240*g0055*p3 *p4  + 120*g0056*p3*p4

                  4               2   3               3   2               2   3
  + 48*g0059*p3*p4  + 120*g0060*p3 *p4  + 120*g0061*p3 *p4  + 240*g0062*p3 *p4

               4                  3   2             5
  + 60*g0063*p3 *p4 + 180*g0064*p3 *p4  + 3*g0065*p3 ,

 p3,

            4                     3                   4                   4
 48*g0007*p4 *q35 + 60*g0008*p3*p4 *q35 + 120*g0009*p4 *q35 + 160*g0010*p4 *q35

                   3                  2   2                  3
  + 240*g0011*p3*p4 *q35 + 40*g0012*p3 *p4 *q35 + 15*g0014*p3 *p4*q35

                2   2                      3                      3
  + 180*g0015*p3 *p4 *q35 + 180*g0016*p3*p4 *q35 + 240*g0017*p3*p4 *q35

                4                  4                      3
  + 120*g0018*p4 *q35 + 48*g0021*p4 *q35 + 120*g0022*p3*p4 *q35

                2   2                      3                  3
  + 120*g0023*p3 *p4 *q35 + 240*g0024*p3*p4 *q35 + 60*g0025*p3 *p4*q35

                2   2                 4                  5
  + 180*g0026*p3 *p4 *q35 + 3*g0027*p3 *q35 + 16*g0039*p4 ,

 q23}

Relevance for the application:

The equation:


f =Df*f *p3 + f  *p4
 t     x       2x

The symmetry:

                   4        15               2   3
f =(3*Df  *f *p3*p4 *q35 + ----*Df  *Df*f *p3 *p4 *q35
 s      4x  x               2     3x     x

        15                4                        2   3
     + ----*Df  *f  *p3*p4 *q35 + 15*Df  *Df *f *p3 *p4 *q35
        2     3x  2x                   2x   x  x

        15           2      3   2                        2   3
     + ----*Df  *(Df) *f *p3 *p4 *q35 + 15*Df  *Df*f  *p3 *p4 *q35
        2     2x        x                    2x     2x

                        4        45       2         3   2
     + 10*Df  *f  *p3*p4 *q35 + ----*(Df ) *Df*f *p3 *p4 *q35
            2x  3x               4      x       x

        45       2       2   3        15          3      4
     + ----*(Df ) *f  *p3 *p4 *q35 + ----*Df *(Df) *f *p3 *p4*q35
        4      x    2x                4     x        x

        45          2       3   2                       2   3
     + ----*Df *(Df) *f  *p3 *p4 *q35 + 15*Df *Df*f  *p3 *p4 *q35
        4     x        2x                    x     3x

        15               4        3       5      5
     + ----*Df *f  *p3*p4 *q35 + ----*(Df) *f *p3 *q35
        2     x  4x               16         x

        15      4       4           5      3       3   2
     + ----*(Df) *f  *p3 *p4*q35 + ---*(Df) *f  *p3 *p4 *q35
        16         2x               2         3x

        15      2       2   3                     4             5        5
     + ----*(Df) *f  *p3 *p4 *q35 + 3*Df*f  *p3*p4 *q35 + f  *p4 *q35)/p4
        4          4x                     5x               6x

And now in machine readable form:

The system:

df(f(1),t)=d(1,f(1))*df(f(1),x)*p3 + df(f(1),x,2)*p4$

The symmetry:

df(f(1),s)=(3*d(1,df(f(1),x,4))*df(f(1),x)*p3*p4**4*q35 + 15/2*d(1,df(f(1),x,3))
*d(1,f(1))*df(f(1),x)*p3**2*p4**3*q35 + 15/2*d(1,df(f(1),x,3))*df(f(1),x,2)*p3*
p4**4*q35 + 15*d(1,df(f(1),x,2))*d(1,df(f(1),x))*df(f(1),x)*p3**2*p4**3*q35 + 15
/2*d(1,df(f(1),x,2))*d(1,f(1))**2*df(f(1),x)*p3**3*p4**2*q35 + 15*d(1,df(f(1),x,
2))*d(1,f(1))*df(f(1),x,2)*p3**2*p4**3*q35 + 10*d(1,df(f(1),x,2))*df(f(1),x,3)*
p3*p4**4*q35 + 45/4*d(1,df(f(1),x))**2*d(1,f(1))*df(f(1),x)*p3**3*p4**2*q35 + 45
/4*d(1,df(f(1),x))**2*df(f(1),x,2)*p3**2*p4**3*q35 + 15/4*d(1,df(f(1),x))*d(1,f(
1))**3*df(f(1),x)*p3**4*p4*q35 + 45/4*d(1,df(f(1),x))*d(1,f(1))**2*df(f(1),x,2)*
p3**3*p4**2*q35 + 15*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,3)*p3**2*p4**3*q35 + 15
/2*d(1,df(f(1),x))*df(f(1),x,4)*p3*p4**4*q35 + 3/16*d(1,f(1))**5*df(f(1),x)*p3**
5*q35 + 15/16*d(1,f(1))**4*df(f(1),x,2)*p3**4*p4*q35 + 5/2*d(1,f(1))**3*df(f(1),
x,3)*p3**3*p4**2*q35 + 15/4*d(1,f(1))**2*df(f(1),x,4)*p3**2*p4**3*q35 + 3*d(1,f(
1))*df(f(1),x,5)*p3*p4**4*q35 + df(f(1),x,6)*p4**5*q35)/p4**5$