Solution 2 to problem N1t10s14f3


Expressions | Parameters | Inequalities | Relevance | Back to problem N1t10s14f3

Expressions

The solution is given through the following expressions:

      3
     ----*p4*q9
      28
q16=------------
         p3


q15=0


     1
q14=---*q9
     2


     5
q13=---*q9
     4


     3
q12=---*q9
     2


      7
     ----*p3*q9
      12
q11=------------
         p4


      7
     ---*p3*q9
      4
q10=-----------
        p4


    1
q8=---*q9
    4


     13
    ----*p3*q9
     12
q7=------------
        p4


     5
    ---*p3*q9
     6
q6=-----------
       p4


     2    2
    ---*p3 *q9
     9
q5=------------
         2
       p4


     1    2
    ---*p3 *q9
     3
q4=------------
         2
       p4


     1
    ---*p3*q9
     2
q3=-----------
       p4


     1
    ---*p3*q9
     4
q2=-----------
       p4


q1=0


     3    2
    ---*p4
     5
p7=---------
      p3


p6=2*p4


p5=3*p4


p2=p3


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:
 q9, 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.
 
{q10,

           2                                                               2
 3*g0042*p4  + 10*g0043*p3*p4 + 15*g0044*p3*p4 + 5*g0045*p3*p4 + 5*g0046*p3

              2
  + 5*g0047*p3 ,

 p2,

 p5,

 p6,

 p4,

 q9,

 q12,

 p7,

 p3,

 q6,

            2                 2                 2
 18*g0006*p4 *q9 + 45*g0007*p4 *q9 + 54*g0008*p4 *q9 + 21*g0009*p3*p4*q9

                                   2                2
  + 63*g0010*p3*p4*q9 + 36*g0011*p4 *q9 + 9*g0012*p4 *q9 + 39*g0013*p3*p4*q9

                                  2                 2
  + 30*g0014*p3*p4*q9 + 8*g0015*p3 *q9 + 12*g0016*p3 *q9 + 18*g0017*p3*p4*q9

                                  3               3              3
  + 9*g0018*p3*p4*q9 + 72*g0020*p4  + 108*g0021*p4  + 36*g0022*p4

                  2                 2
  + 36*g0023*p3*p4  + 36*g0024*p3*p4 ,

 q7,

            3                  2                  2                  2
 27*g0026*p4  + 126*g0028*p3*p4  + 315*g0029*p3*p4  + 378*g0030*p3*p4

                2                  2                     2                 2
  + 147*g0031*p3 *p4 + 441*g0032*p3 *p4 + 252*g0033*p3*p4  + 63*g0034*p3*p4

                2                  2                 3              3
  + 273*g0035*p3 *p4 + 210*g0036*p3 *p4 + 56*g0037*p3  + 84*g0038*p3

                2                 2
  + 126*g0039*p3 *p4 + 63*g0040*p3 *p4}


Relevance for the application:



The equation: 


                               2                         2      2
f =(Df  *f *p3*p4 + Df *Df*f*p3  + 3*Df *f  *p3*p4 + (Df) *f *p3
 t    2x  x           x                x  2x                x

                         3        2
     + 2*Df*f  *p3*p4 + ---*f  *p4 )/p3
             3x          5   5x
The symmetry:
     1               2       1              2                       2
f =(---*Df  *f *p3*p4 *q9 + ---*Df  *Df*f*p3 *p4*q9 + Df  *f  *p3*p4 *q9
 s   4    4x  x              4    3x                    3x  2x

        1               2          13               2
     + ---*Df  *Df *f*p3 *p4*q9 + ----*Df  *Df*f *p3 *p4*q9
        2    2x   x                12    2x     x

        3                2       5       2      2          1          2     3
     + ---*Df  *f  *p3*p4 *q9 + ---*(Df ) *f *p3 *p4*q9 + ---*Df *(Df) *f*p3 *q9
        2    2x  3x              6     x    x              3    x

        7               2          5               2       2      3      3
     + ---*Df *Df*f  *p3 *p4*q9 + ---*Df *f  *p3*p4 *q9 + ---*(Df) *f *p3 *q9
        4    x     2x              4    x  4x              9         x

        7       2       2          1              2       3         3
     + ----*(Df) *f  *p3 *p4*q9 + ---*Df*f  *p3*p4 *q9 + ----*f  *p4 *q9)/(p3
        12         3x              2      5x              28   7x

      2
   *p4 )
And now in machine readable form:

The system:

df(f(1),t)=(d(1,df(f(1),x,2))*df(f(1),x)*p3*p4 + d(1,df(f(1),x))*d(1,f(1))*f(1)*
p3**2 + 3*d(1,df(f(1),x))*df(f(1),x,2)*p3*p4 + d(1,f(1))**2*df(f(1),x)*p3**2 + 2
*d(1,f(1))*df(f(1),x,3)*p3*p4 + 3/5*df(f(1),x,5)*p4**2)/p3$
The symmetry:
df(f(1),s)=(1/4*d(1,df(f(1),x,4))*df(f(1),x)*p3*p4**2*q9 + 1/4*d(1,df(f(1),x,3))
*d(1,f(1))*f(1)*p3**2*p4*q9 + d(1,df(f(1),x,3))*df(f(1),x,2)*p3*p4**2*q9 + 1/2*d
(1,df(f(1),x,2))*d(1,df(f(1),x))*f(1)*p3**2*p4*q9 + 13/12*d(1,df(f(1),x,2))*d(1,
f(1))*df(f(1),x)*p3**2*p4*q9 + 3/2*d(1,df(f(1),x,2))*df(f(1),x,3)*p3*p4**2*q9 + 
5/6*d(1,df(f(1),x))**2*df(f(1),x)*p3**2*p4*q9 + 1/3*d(1,df(f(1),x))*d(1,f(1))**2
*f(1)*p3**3*q9 + 7/4*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*p3**2*p4*q9 + 5/4*d(
1,df(f(1),x))*df(f(1),x,4)*p3*p4**2*q9 + 2/9*d(1,f(1))**3*df(f(1),x)*p3**3*q9 + 
7/12*d(1,f(1))**2*df(f(1),x,3)*p3**2*p4*q9 + 1/2*d(1,f(1))*df(f(1),x,5)*p3*p4**2
*q9 + 3/28*df(f(1),x,7)*p4**3*q9)/(p3*p4**2)$