Solution 11 to problem N1t8s14f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t8s14f1

Expressions

The solution is given through the following expressions:


q55=0


q54=0


q53=0


q52=0


q51=0


q50=0


q49=0


q48=0


q47=0


q46=0


q45=6*q3


q44=0


q43=0


q42=0


q41=0


q40=0


q39= - 18*q3 + 5*q6


q38=5*q5


q37=2*q3


q36=0


q35=0


q34=0


q33=0


q32=0


q31=0


q30=0


q29=4*q8


q28=4*q10 - 14*q5


q27= - 12*q3 + 5*q6


q26=0


q25=0


q24= - 3*q3


q23=0


q22=0


q21= - 3*q5


q20=6*q3 - 3*q6


q19=4*q3 - q6


q18= - 3*q8


        3         9
q17= - ---*q10 + ---*q5
        2         2


        1
q16= - ---*q12
        2


q15=0


q14=q10 - 3*q5


q13=4*q8


q11= - 6*q3 + 2*q6


q9= - 2*q3 + q6


q7=0


q4=0


q2=0


q1=0


p13=0


p12=0


p11=0


p10=0


p9=0


p8=0


p7=0


p5=0


p4= - p6


p3=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:

 q12, q10, q8, q6, q5, q3, p6

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.

 
{12*g0081*q3 - 36*g0087*q3 + 10*g0087*q6 + 10*g0088*q5 + 4*g0089*q3 + 8*g0097*q8

  + 8*g0098*q10 - 28*g0098*q5 - 24*g0099*q3 + 10*g0099*q6 - 6*g0102*q3

  - 6*g0105*q5 + 12*g0106*q3 - 6*g0106*q6 + 8*g0107*q3 - 2*g0107*q6 - 6*g0108*q8

  - 3*g0109*q10 + 9*g0109*q5 - g0110*q12 + 2*g0112*q10 - 6*g0112*q5 + 8*g0113*q8

  + 2*g0114*q12 - 12*g0115*q3 + 4*g0115*q6 + 2*g0116*q10 - 4*g0117*q3

  + 2*g0117*q6 + 2*g0118*q8 + 2*g0120*q6 + 2*g0121*q5 + 2*g0123*q3,

 12*g0014*q3 - 36*g0020*q3 + 10*g0020*q6 + 10*g0021*q5 + 4*g0022*q3 + 8*g0030*q8

  + 8*g0031*q10 - 28*g0031*q5 - 24*g0032*q3 + 10*g0032*q6 - 6*g0035*q3

  - 6*g0038*q5 + 12*g0039*q3 - 6*g0039*q6 + 8*g0040*q3 - 2*g0040*q6 - 6*g0041*q8

  - 3*g0042*q10 + 9*g0042*q5 - g0043*q12 + 2*g0045*q10 - 6*g0045*q5 + 8*g0046*q8

  + 2*g0047*q12 - 12*g0048*q3 + 4*g0048*q6 + 2*g0049*q10 - 4*g0050*q3

  + 2*g0050*q6 + 2*g0051*q8 + 2*g0053*q6 + 2*g0054*q5 + 2*g0056*q3 + 2*g0065*p6

  - 2*g0067*p6,

 p6,

 p4}

Relevance for the application:

The equation:


              2            3
f = - Df *(Df) *f*p6 + (Df) *f *p6
 t      x                     x

The symmetry:

            2                                    3                   2
f =Df  *(Df) *f*q3 + Df  *Df *Df*f*q6 + Df  *(Df) *f*q5 - 3*Df  *(Df) *f *q3
 s   4x                3x   x             3x                  3x        x

           2                 2                       2                    2
 - 2*(Df  ) *Df*f*q3 + (Df  ) *Df*f*q6 - 6*Df  *(Df ) *f*q3 + 2*Df  *(Df ) *f*q6
        2x                2x                 2x    x              2x    x

                2
 + Df  *Df *(Df) *f*q10 + 6*Df  *Df *Df*f *q3 - 3*Df  *Df *Df*f *q6
     2x   x                   2x   x     x          2x   x     x

            4                   3
 + Df  *(Df) *f*q8 - 3*Df  *(Df) *f *q5 - 12*Df  *f  *f *f*q3
     2x                  2x        x           2x  2x  x

                             3                   3                  3
 + 5*Df  *f  *f *f*q6 + (Df ) *Df*f*q10 - 3*(Df ) *Df*f*q5 + 4*(Df ) *f *q3
       2x  2x  x           x                   x                  x    x

        3                2     3         3       2     2
 - (Df ) *f *q6 + 4*(Df ) *(Df) *f*q8 - ---*(Df ) *(Df) *f *q10
      x    x           x                 2     x          x

    9       2     2                 5                   4
 + ---*(Df ) *(Df) *f *q5 + Df *(Df) *f*q12 - 3*Df *(Df) *f *q8
    2     x          x        x                   x        x

 + 4*Df *Df*f  *f *f*q10 - 14*Df *Df*f  *f *f*q5 - 18*Df *f  *f *f*q3
       x     2x  x              x     2x  x             x  3x  x

                        1      6                3
 + 5*Df *f  *f *f*q6 - ---*(Df) *f *q12 + 4*(Df) *f  *f *f*q8
       x  3x  x         2         x                2x  x

         2
 + 5*(Df) *f  *f *f*q5 + 6*Df*f  *f *f*q3 + 2*Df*f  *f  *f*q3
            3x  x              4x  x              3x  2x

And now in machine readable form:

The system:

df(f(1),t)= - d(1,df(f(1),x))*d(1,f(1))**2*f(1)*p6 + d(1,f(1))**3*df(f(1),x)*p6$

The symmetry:

df(f(1),s)=d(1,df(f(1),x,4))*d(1,f(1))**2*f(1)*q3 + d(1,df(f(1),x,3))*d(1,df(f(1
),x))*d(1,f(1))*f(1)*q6 + d(1,df(f(1),x,3))*d(1,f(1))**3*f(1)*q5 - 3*d(1,df(f(1)
,x,3))*d(1,f(1))**2*df(f(1),x)*q3 - 2*d(1,df(f(1),x,2))**2*d(1,f(1))*f(1)*q3 + d
(1,df(f(1),x,2))**2*d(1,f(1))*f(1)*q6 - 6*d(1,df(f(1),x,2))*d(1,df(f(1),x))**2*f
(1)*q3 + 2*d(1,df(f(1),x,2))*d(1,df(f(1),x))**2*f(1)*q6 + d(1,df(f(1),x,2))*d(1,
df(f(1),x))*d(1,f(1))**2*f(1)*q10 + 6*d(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1)
)*df(f(1),x)*q3 - 3*d(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x)*q6 + 
d(1,df(f(1),x,2))*d(1,f(1))**4*f(1)*q8 - 3*d(1,df(f(1),x,2))*d(1,f(1))**3*df(f(1
),x)*q5 - 12*d(1,df(f(1),x,2))*df(f(1),x,2)*df(f(1),x)*f(1)*q3 + 5*d(1,df(f(1),x
,2))*df(f(1),x,2)*df(f(1),x)*f(1)*q6 + d(1,df(f(1),x))**3*d(1,f(1))*f(1)*q10 - 3
*d(1,df(f(1),x))**3*d(1,f(1))*f(1)*q5 + 4*d(1,df(f(1),x))**3*df(f(1),x)*q3 - d(1
,df(f(1),x))**3*df(f(1),x)*q6 + 4*d(1,df(f(1),x))**2*d(1,f(1))**3*f(1)*q8 - 3/2*
d(1,df(f(1),x))**2*d(1,f(1))**2*df(f(1),x)*q10 + 9/2*d(1,df(f(1),x))**2*d(1,f(1)
)**2*df(f(1),x)*q5 + d(1,df(f(1),x))*d(1,f(1))**5*f(1)*q12 - 3*d(1,df(f(1),x))*d
(1,f(1))**4*df(f(1),x)*q8 + 4*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*df(f(1),x)*
f(1)*q10 - 14*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*df(f(1),x)*f(1)*q5 - 18*d(1
,df(f(1),x))*df(f(1),x,3)*df(f(1),x)*f(1)*q3 + 5*d(1,df(f(1),x))*df(f(1),x,3)*df
(f(1),x)*f(1)*q6 - 1/2*d(1,f(1))**6*df(f(1),x)*q12 + 4*d(1,f(1))**3*df(f(1),x,2)
*df(f(1),x)*f(1)*q8 + 5*d(1,f(1))**2*df(f(1),x,3)*df(f(1),x)*f(1)*q5 + 6*d(1,f(1
))*df(f(1),x,4)*df(f(1),x)*f(1)*q3 + 2*d(1,f(1))*df(f(1),x,3)*df(f(1),x,2)*f(1)*
q3$