Solution 3 to problem N1t8s12f1

Expressions | Parameters | Inequalities | Relevance | Back to problem N1t8s12f1

Expressions

The solution is given through the following expressions:


q35=0


q34=0


q33=0


q32=0


q31=0


q30=0


q29=0


q28=0


      - 24*p10*q27 + 9*p12*q27
q26=---------------------------
               4*p10


q25=0


q24=0


q23=0


q22=0


q21=0


q20=0


q18=0


q17=0


     12*p10*q27 - 3*p12*q27
q16=------------------------
             4*p10


q15=0


q14=0


     24*p10*q27 - 3*p12*q27
q13=------------------------
             4*p10


q12=0


q11=0


q10=0


q9=0


     - 24*p10*q27 + 3*p12*q27
q8=---------------------------
              4*p10


     - 18*p10*q27 + 3*p12*q27
q7=---------------------------
               p10


q6=0


q5=0


q4=0


     - 24*p10*q27 + 3*p12*q27
q3=---------------------------
              4*p10


q2=0


q1=0


p13=0


p11=0


p9=0


p8=0


    4*p10 - p12
p7=-------------
         2


p6=0


p5=0


p4=0


     - 4*p10 + p12
p3=----------------
          2


p1=0


     - 8*p10 + p12
p2=----------------
          2


      - 72*p10*q27 + 9*p12*q27
q19=---------------------------
               4*p10

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:

 q27, p12, p10

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.

 
{2*g0059*p10 + 24*g0060*p10 + 9*g0060*p2 + 9*g0067*p2 - 6*g0070*p10 - 3*g0070*p2

  - 3*g0073*p2 + 3*g0078*p2 + 12*g0079*p10 + 12*g0079*p2 + 3*g0083*p2,

 p10 + p2,

 p10,

 8*g0087*p10 + 2*g0087*p2 + g0089*p10 - 2*g0092*p10 - g0092*p2 + 2*g0096*p10

  + g0096*p2 + g0097*p2}

Relevance for the application:

The equation:


                         1                         2          1       2
f = - 4*Df  *Df*f*p10 + ---*Df  *Df*f*p12 - 2*(Df ) *f*p10 + ---*(Df ) *f*p12
 t        2x             2    2x                 x            2     x

                      1                      2
 + 2*Df *Df*f *p10 - ---*Df *Df*f *p12 + (Df) *f  *p10 + f  *f *f*p12
       x     x        2    x     x              2x        2x  x

The symmetry:

                  2              3           2
f =( - 6*Df  *(Df) *f*p10*q27 + ---*Df  *(Df) *f*p12*q27
 s         3x                    4    3x

     - 18*Df  *Df *Df*f*p10*q27 + 3*Df  *Df *Df*f*p12*q27
            2x   x                    2x   x

                  2               3           2                     3
     + 3*Df  *(Df) *f *p10*q27 - ---*Df  *(Df) *f *p12*q27 - 6*(Df ) *f*p10*q27
           2x        x            4    2x        x                x

        3       3                    2                  3       2
     + ---*(Df ) *f*p12*q27 + 6*(Df ) *Df*f *p10*q27 - ---*(Df ) *Df*f *p12*q27
        4     x                    x       x            4     x       x

                                  9                             3
     - 18*Df *f  *f *f*p10*q27 + ---*Df *f  *f *f*p12*q27 + (Df) *f  *p10*q27
            x  2x  x              4    x  2x  x                    3x

                                9
     - 6*Df*f  *f *f*p10*q27 + ---*Df*f  *f *f*p12*q27)/p10
             3x  x              4      3x  x

And now in machine readable form:

The system:

df(f(1),t)= - 4*d(1,df(f(1),x,2))*d(1,f(1))*f(1)*p10 + 1/2*d(1,df(f(1),x,2))*d(1
,f(1))*f(1)*p12 - 2*d(1,df(f(1),x))**2*f(1)*p10 + 1/2*d(1,df(f(1),x))**2*f(1)*
p12 + 2*d(1,df(f(1),x))*d(1,f(1))*df(f(1),x)*p10 - 1/2*d(1,df(f(1),x))*d(1,f(1))
*df(f(1),x)*p12 + d(1,f(1))**2*df(f(1),x,2)*p10 + df(f(1),x,2)*df(f(1),x)*f(1)*
p12$

The symmetry:

df(f(1),s)=( - 6*d(1,df(f(1),x,3))*d(1,f(1))**2*f(1)*p10*q27 + 3/4*d(1,df(f(1),x
,3))*d(1,f(1))**2*f(1)*p12*q27 - 18*d(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1))*
f(1)*p10*q27 + 3*d(1,df(f(1),x,2))*d(1,df(f(1),x))*d(1,f(1))*f(1)*p12*q27 + 3*d(
1,df(f(1),x,2))*d(1,f(1))**2*df(f(1),x)*p10*q27 - 3/4*d(1,df(f(1),x,2))*d(1,f(1)
)**2*df(f(1),x)*p12*q27 - 6*d(1,df(f(1),x))**3*f(1)*p10*q27 + 3/4*d(1,df(f(1),x)
)**3*f(1)*p12*q27 + 6*d(1,df(f(1),x))**2*d(1,f(1))*df(f(1),x)*p10*q27 - 3/4*d(1,
df(f(1),x))**2*d(1,f(1))*df(f(1),x)*p12*q27 - 18*d(1,df(f(1),x))*df(f(1),x,2)*df
(f(1),x)*f(1)*p10*q27 + 9/4*d(1,df(f(1),x))*df(f(1),x,2)*df(f(1),x)*f(1)*p12*q27
 + d(1,f(1))**3*df(f(1),x,3)*p10*q27 - 6*d(1,f(1))*df(f(1),x,3)*df(f(1),x)*f(1)*
p10*q27 + 9/4*d(1,f(1))*df(f(1),x,3)*df(f(1),x)*f(1)*p12*q27)/p10$