Solution 1 to problem N1t10s14f3

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

Expressions

The solution is given through the following expressions:


      25    2
     ----*p7 *q7
      91
q16=-------------
           2
         p1


     1
q15=----*q7
     13


      5
     ----*p7*q7
      13
q14=------------
         p1


      15
     ----*p7*q7
      13
q13=------------
         p1


      25
     ----*p7*q7
      13
q12=------------
         p1


     3
q11=----*q7
     13


     12
q10=----*q7
     13


     25
    ----*p7*q7
     13
q9=------------
        p1


     15
    ----*p7*q7
     13
q8=------------
        p1


    10
q6=----*q7
    13


     1
    ----*p1*q7
     13
q5=------------
        p7


     3
    ----*p1*q7
     13
q4=------------
        p7


    15
q3=----*q7
    13


    7
q2=----*q7
    13


     5
    ----*p7*q7
     13
q1=------------
        p1


p6=p1


p5=2*p1


p4=2*p1


     2    2
    ---*p1
     5
p3=---------
      p7


     4    2
    ---*p1
     5
p2=---------
      p7

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:

 q7, p7, p1

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.

 
{p3,

           2                                                               2
 5*g0042*p7  + 5*g0043*p1*p7 + 10*g0044*p1*p7 + 10*g0045*p1*p7 + 2*g0046*p1

              2
  + 4*g0047*p1  + 5*g0048*p1*p7,

 q10,

 q2,

 p2,

 p1,

 p7,

 q1,

 q3,

 q7,

 q6,

                               2                 2                  2
 5*g0005*p1*p7*q7 + 25*g0006*p7 *q7 + 75*g0007*p7 *q7 + 125*g0008*p7 *q7

                                                        2                 2
  + 15*g0009*p1*p7*q7 + 60*g0010*p1*p7*q7 + 125*g0011*p7 *q7 + 75*g0012*p7 *q7

                                                      2                 2
  + 65*g0013*p1*p7*q7 + 50*g0014*p1*p7*q7 + 5*g0015*p1 *q7 + 15*g0016*p1 *q7

                                                       2                 2
  + 75*g0017*p1*p7*q7 + 35*g0018*p1*p7*q7 + 25*g0019*p7 *q7 + 65*g0020*p1 *p7

                2                  2                 3              3
  + 130*g0021*p1 *p7 + 130*g0022*p1 *p7 + 26*g0023*p1  + 52*g0024*p1

               2
  + 65*g0025*p1 *p7,

 q4,

            3             2                    2                  2
 25*g0026*p7  + 7*g0027*p1 *p7 + 35*g0028*p1*p7  + 105*g0029*p1*p7

                   2              2                 2                     2
  + 175*g0030*p1*p7  + 21*g0031*p1 *p7 + 84*g0032*p1 *p7 + 175*g0033*p1*p7

                   2              2                 2                3
  + 105*g0034*p1*p7  + 91*g0035*p1 *p7 + 70*g0036*p1 *p7 + 7*g0037*p1

               3               2                 2                    2
  + 21*g0038*p1  + 105*g0039*p1 *p7 + 49*g0040*p1 *p7 + 35*g0041*p1*p7 }

Relevance for the application:

The equation:


                                      4             2
f =(Df  *f*p1*p7 + 2*Df  *f *p1*p7 + ---*Df *Df*f*p1  + 2*Df *f  *p1*p7
 t    3x               2x  x          5    x                x  2x

        2      2      2                        2
     + ---*(Df) *f *p1  + Df*f  *p1*p7 + f  *p7 )/p7
        5         x           3x          5x

The symmetry:

     5               2       15               2       7               2
f =(----*Df  *f*p1*p7 *q7 + ----*Df  *f *p1*p7 *q7 + ----*Df  *Df*f*p1 *p7*q7
 s   13    5x                13    4x  x              13    3x

        25                2       15               2
     + ----*Df  *f  *p1*p7 *q7 + ----*Df  *Df *f*p1 *p7*q7
        13    3x  2x              13    2x   x

                    2          25                2       10       2      2
     + Df  *Df*f *p1 *p7*q7 + ----*Df  *f  *p1*p7 *q7 + ----*(Df ) *f *p1 *p7*q7
         2x     x              13    2x  3x              13     x    x

        3           2     3       12               2
     + ----*Df *(Df) *f*p1 *q7 + ----*Df *Df*f  *p1 *p7*q7
        13    x                   13    x     2x

        15               2       1       3      3       3       2       2
     + ----*Df *f  *p1*p7 *q7 + ----*(Df) *f *p1 *q7 + ----*(Df) *f  *p1 *p7*q7
        13    x  4x              13         x           13         3x

        5               2       25        3       1              2           2
     + ----*Df*f  *p1*p7 *q7 + ----*f  *p7 *q7 + ----*f  *f *f*p1 *p7*q7)/(p1
        13      5x              91   7x           13   3x  x

   *p7)

And now in machine readable form:

The system:

df(f(1),t)=(d(1,df(f(1),x,3))*f(1)*p1*p7 + 2*d(1,df(f(1),x,2))*df(f(1),x)*p1*p7 
+ 4/5*d(1,df(f(1),x))*d(1,f(1))*f(1)*p1**2 + 2*d(1,df(f(1),x))*df(f(1),x,2)*p1*
p7 + 2/5*d(1,f(1))**2*df(f(1),x)*p1**2 + d(1,f(1))*df(f(1),x,3)*p1*p7 + df(f(1),
x,5)*p7**2)/p7$

The symmetry:

df(f(1),s)=(5/13*d(1,df(f(1),x,5))*f(1)*p1*p7**2*q7 + 15/13*d(1,df(f(1),x,4))*df
(f(1),x)*p1*p7**2*q7 + 7/13*d(1,df(f(1),x,3))*d(1,f(1))*f(1)*p1**2*p7*q7 + 25/13
*d(1,df(f(1),x,3))*df(f(1),x,2)*p1*p7**2*q7 + 15/13*d(1,df(f(1),x,2))*d(1,df(f(1
),x))*f(1)*p1**2*p7*q7 + d(1,df(f(1),x,2))*d(1,f(1))*df(f(1),x)*p1**2*p7*q7 + 25
/13*d(1,df(f(1),x,2))*df(f(1),x,3)*p1*p7**2*q7 + 10/13*d(1,df(f(1),x))**2*df(f(1
),x)*p1**2*p7*q7 + 3/13*d(1,df(f(1),x))*d(1,f(1))**2*f(1)*p1**3*q7 + 12/13*d(1,
df(f(1),x))*d(1,f(1))*df(f(1),x,2)*p1**2*p7*q7 + 15/13*d(1,df(f(1),x))*df(f(1),x
,4)*p1*p7**2*q7 + 1/13*d(1,f(1))**3*df(f(1),x)*p1**3*q7 + 3/13*d(1,f(1))**2*df(f
(1),x,3)*p1**2*p7*q7 + 5/13*d(1,f(1))*df(f(1),x,5)*p1*p7**2*q7 + 25/91*df(f(1),x
,7)*p7**3*q7 + 1/13*df(f(1),x,3)*df(f(1),x)*f(1)*p1**2*p7*q7)/(p1**2*p7)$