Solution 3 to problem N1t4s14f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t4s14f1

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=4*q3


q44=0


q43=0


q42=0


q41=0


q40=0


q39= - 10*q3 + 3*q6


q38=3*q5


q37=2*q3


q36=0


q35=0


q34=0


q33=0


q32=0


q31=0


q30=0


q29=2*q8


q28=2*q10 - 6*q5


q27= - 8*q3 + 3*q6


q26=0


q25=0


q24= - q3


q23=0


q22=0


q21= - q5


q20=2*q3 - q6


     4        1
q19=---*q3 - ---*q6
     3        3


q18= - q8


        1         3
q17= - ---*q10 + ---*q5
        2         2


        1
q16= - ---*q12
        6


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


p4=0


p2=0


p1= - p3

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, p3

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. 
 
{p1,

 p3,

 24*g0014*q3 - 60*g0020*q3 + 18*g0020*q6 + 18*g0021*q5 + 12*g0022*q3

  + 12*g0030*q8 + 12*g0031*q10 - 36*g0031*q5 - 48*g0032*q3 + 18*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 + 6*g0045*q10

  - 18*g0045*q5 + 24*g0046*q8 + 6*g0047*q12 - 36*g0048*q3 + 12*g0048*q6

  + 6*g0049*q10 - 12*g0050*q3 + 6*g0050*q6 + 6*g0051*q8 + 6*g0053*q6

  + 6*g0054*q5 + 6*g0056*q3 + 6*g0059*p3 - 6*g0061*p3,

 24*g0072*q3 - 60*g0078*q3 + 18*g0078*q6 + 18*g0079*q5 + 12*g0080*q3

  + 12*g0088*q8 + 12*g0089*q10 - 36*g0089*q5 - 48*g0090*q3 + 18*g0090*q6

  - 6*g0093*q3 - 6*g0096*q5 + 12*g0097*q3 - 6*g0097*q6 + 8*g0098*q3 - 2*g0098*q6

  - 6*g0099*q8 - 3*g0100*q10 + 9*g0100*q5 - g0101*q12 + 6*g0103*q10

  - 18*g0103*q5 + 24*g0104*q8 + 6*g0105*q12 - 36*g0106*q3 + 12*g0106*q6

  + 6*g0107*q10 - 12*g0108*q3 + 6*g0108*q6 + 6*g0109*q8 + 6*g0111*q6

  + 6*g0112*q5 + 6*g0114*q3}

Relevance for the application:



The equation: 



f = - Df *f*p3 + Df*f *p3
 t      x            x


The symmetry: 

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

            4                 3
 + Df  *(Df) *f*q8 - Df  *(Df) *f *q5 - 8*Df  *f  *f *f*q3 + 3*Df  *f  *f *f*q6
     2x                2x        x          2x  2x  x            2x  2x  x

        3                   3            4       3          1       3
 + (Df ) *Df*f*q10 - 3*(Df ) *Df*f*q5 + ---*(Df ) *f *q3 - ---*(Df ) *f *q6
      x                   x              3     x    x       3     x    x

          2     3         1       2     2           3       2     2
 + 4*(Df ) *(Df) *f*q8 - ---*(Df ) *(Df) *f *q10 + ---*(Df ) *(Df) *f *q5
        x                 2     x          x        2     x          x

           5                 4
 + Df *(Df) *f*q12 - Df *(Df) *f *q8 + 2*Df *Df*f  *f *f*q10
     x                 x        x          x     2x  x

 - 6*Df *Df*f  *f *f*q5 - 10*Df *f  *f *f*q3 + 3*Df *f  *f *f*q6
       x     2x  x             x  3x  x            x  3x  x

    1      6                3                     2
 - ---*(Df) *f *q12 + 2*(Df) *f  *f *f*q8 + 3*(Df) *f  *f *f*q5
    6         x                2x  x                 3x  x

 + 4*Df*f  *f *f*q3 + 2*Df*f  *f  *f*q3
         4x  x              3x  2x


And now in machine readable form: 

The system: 

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


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 - 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 + 2*d(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1))*
df(f(1),x)*q3 - 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 - d(1,df(f(1),x,2))*d(1,f(1))**3*df(f(1),x)*
q5 - 8*d(1,df(f(1),x,2))*df(f(1),x,2)*df(f(1),x)*f(1)*q3 + 3*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 + 4/3*d(1,df(f(1),x))**3*df(f(1),x)*q3 - 1/3*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 - 1/2
*d(1,df(f(1),x))**2*d(1,f(1))**2*df(f(1),x)*q10 + 3/2*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 - d(1,df(f(1),x))*d(
1,f(1))**4*df(f(1),x)*q8 + 2*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*df(f(1),x)*f
(1)*q10 - 6*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*df(f(1),x)*f(1)*q5 - 10*d(1,
df(f(1),x))*df(f(1),x,3)*df(f(1),x)*f(1)*q3 + 3*d(1,df(f(1),x))*df(f(1),x,3)*df(
f(1),x)*f(1)*q6 - 1/6*d(1,f(1))**6*df(f(1),x)*q12 + 2*d(1,f(1))**3*df(f(1),x,2)*
df(f(1),x)*f(1)*q8 + 3*d(1,f(1))**2*df(f(1),x,3)*df(f(1),x)*f(1)*q5 + 4*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$