Solution 1 to problem N2t2s14f3

Expressions | Parameters | Inequalities | Relevance | Back to problem N2t2s14f3

Expressions

The solution is given through the following expressions:


q81=0


q80=0


q79=0


q78=0


q77=0


q76=0


q75=0


q74=q28


q73=q29


q72= - q29


q71=q28


q70=q45


q69=q46


q68=q46


q67= - q45


q66= - q60


q65=q61


        1
q64= - ---*q24
        2


q63= - q25


     1
q62=---*q24
     2


        1
q59= - ---*q25
        2


q58=q24


     1
q57=---*q25
     2


q56= - 2*q21


q55=q22


q54= - q22


q53= - 2*q21


q52= - q45


q51=q46


q50=q22


q49=2*q21


q48= - 2*q21


q47=q22


q44=q28


q43= - q29


q42= - q25


q41= - q24


q40=q24


q39= - q25


        1
q38= - ---*q22
        2


q37= - 2*q21


     1
q36=---*q22
     2


q35=q17


q34= - 3*q17


q33= - q19


     1
q32=---*q19
     3


q31=0


q30=0


q27=q24


q26= - q25


q23= - q21


        1
q20= - ---*q19
        3


q18= - 3*q17


q16=0


q15=0


q14=0


q13=0


q12=0


q11=0


q10=0


q9=0


q8=0


q7=0


q6=0


q5=0


q4=0


q3=0


q2=0


q1=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:

 q83, q82, q61, q60, q46, q45, q29, q28, q25, q24, q22, 
q21, q19, q17, p2, 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.

 
{p1,

 g0220*p2 + g0221*p1,

 6*g0003*q82 + 6*g0010*q28 + 6*g0011*q29 + 6*g0012*q28 + 6*g0013*q45

  + 6*g0014*q46 + 6*g0015*q46 + 6*g0016*q61 - 6*g0017*q25 + 3*g0018*q24

  + 6*g0019*q61 + 6*g0020*q60 - 3*g0021*q25 + 6*g0022*q24 + 3*g0023*q25

  - 12*g0024*q21 + 6*g0025*q22 - 12*g0026*q21 + 6*g0027*q46 + 6*g0028*q22

  + 12*g0029*q21 - 12*g0030*q21 + 6*g0031*q22 + 6*g0032*q46 + 6*g0033*q45

  + 6*g0034*q28 - 6*g0035*q25 + 6*g0036*q24 - 6*g0037*q25 - 12*g0038*q21

  + 3*g0039*q22 - 18*g0040*q17 - 6*g0041*q19 + 2*g0042*q19 + 6*g0045*q29

  + 6*g0046*q28 + 6*g0047*q24 - 6*g0048*q25 + 6*g0049*q25 + 6*g0050*q24

  - 6*g0051*q21 + 6*g0052*q22 + 6*g0053*q21 - 2*g0054*q19 + 6*g0055*q19

  - 18*g0056*q17 + 6*g0057*q17 + 6*g0070*p1,

 6*g0071*q82 + 6*g0078*q28 + 6*g0079*q29 - 6*g0080*q29 + 6*g0081*q45

  + 6*g0082*q46 - 6*g0083*q45 - 6*g0084*q60 - 3*g0085*q24 - 6*g0086*q25

  + 6*g0087*q61 + 6*g0088*q60 - 3*g0089*q25 + 6*g0090*q24 + 3*g0091*q25

  - 12*g0092*q21 - 6*g0093*q22 - 12*g0094*q21 - 6*g0095*q45 + 6*g0096*q22

  + 12*g0097*q21 - 12*g0098*q21 + 6*g0099*q22 + 6*g0100*q46 + 6*g0101*q45

  - 6*g0102*q29 - 6*g0103*q25 - 6*g0104*q24 - 6*g0105*q25 - 3*g0106*q22

  - 12*g0107*q21 + 6*g0108*q17 - 18*g0109*q17 - 6*g0110*q19 + 6*g0113*q29

  + 6*g0114*q28 + 6*g0115*q24 - 6*g0116*q25 + 6*g0117*q25 + 6*g0118*q24

  - 6*g0119*q21 + 6*g0120*q22 + 6*g0121*q21 - 2*g0122*q19 + 6*g0123*q19

  - 18*g0124*q17 + 6*g0125*q17 + 6*g0138*p1,

 6*g0146*q28 + 6*g0147*q29 - 6*g0148*q29 + 6*g0149*q28 + 6*g0150*q45

  + 6*g0151*q46 + 6*g0152*q46 - 6*g0153*q45 - 6*g0154*q60 + 6*g0155*q61

  - 3*g0156*q24 - 6*g0157*q25 + 3*g0158*q24 + 6*g0159*q61 + 6*g0160*q60

  - 3*g0161*q25 + 6*g0162*q24 + 3*g0163*q25 - 12*g0164*q21 + 6*g0165*q22

  - 6*g0166*q22 - 12*g0167*q21 - 6*g0168*q45 + 6*g0169*q46 + 6*g0170*q22

  + 12*g0171*q21 - 12*g0172*q21 + 6*g0173*q22 + 6*g0174*q46 + 6*g0175*q45

  + 6*g0176*q28 - 6*g0177*q29 - 6*g0178*q25 - 6*g0179*q24 + 6*g0180*q24

  - 6*g0181*q25 - 3*g0182*q22 - 12*g0183*q21 + 3*g0184*q22 + 6*g0185*q17

  - 18*g0186*q17 - 6*g0187*q19 + 2*g0188*q19 + 6*g0191*q29 + 6*g0192*q28

  + 6*g0193*q24 - 6*g0194*q25 + 6*g0195*q25 + 6*g0196*q24 - 6*g0197*q21

  + 6*g0198*q22 + 6*g0199*q21 - 2*g0200*q19 + 6*g0201*q19 - 18*g0202*q17

  + 6*g0203*q17}

Relevance for the application:

The equation:


f =D D f*p1 + f *p2
 t  1 2        x

The symmetry:

f =D f  *D D f*q29 + D f  *f *q28 + D f  *D D f *q45 + D f  *f  *q46
 s  2 4x  1 2         2 4x  x        2 3x  1 2 x        2 3x  2x

 + D f  *D f*D D f*q25 + D f  *D f*f *q24 + D f  *D D f  *q60
    2 2x  2   1 2         2 2x  2   x        2 2x  1 2 2x

                                                                  2
 + D f  *D D f*D f*q24 - D f  *D f*f *q25 + D f  *f  *q61 + (D f ) *D D f*q21
    2 2x  1 2   1         2 2x  1   x        2 2x  3x         2 x    1 2

    1        2
 + ---*(D f ) *f *q22 + 2*D f *D f*D D f *q21 + D f *D f*f  *q22
    2    2 x    x          2 x  2   1 2 x        2 x  2   2x

 + D f *D D f  *q45 + D f *D D f *D f*q22 + D f *D D f*D f *q22
    2 x  1 2 3x        2 x  1 2 x  1         2 x  1 2   1 x

                                                                 3
 - 2*D f *D f *f *q21 - 2*D f *D f*f  *q21 + D f *f  *q46 + (D f) *D D f*q17
      2 x  1 x  x          2 x  1   2x        2 x  4x         2     1 2

    1       3           1       2                    2
 + ---*(D f) *f *q19 + ---*(D f) *D D f  *q25 + (D f) *D D f*D f*q19
    3    2     x        2    2     1 2 2x         2     1 2   1

          2               1       2
 - 3*(D f) *D f*f *q17 + ---*(D f) *f  *q24 + D f*D D f  *q29
       2     1   x        2    2     3x        2   1 2 4x

 + D f*D D f  *D f*q24 + D f*D D f *D f *q22 + D f*D D f*D f  *q24
    2   1 2 2x  1         2   1 2 x  1 x        2   1 2   1 2x

                    2
 - 3*D f*D D f*(D f) *q17 - D f*D f  *f *q25 - 2*D f*D f *f  *q21
      2   1 2    1           2   1 2x  x          2   1 x  2x

            2
 - D f*(D f) *f *q19 - D f*D f*f  *q25 + D f*f  *q28 + D D f  *q82
    2    1     x        2   1   3x        2   5x        1 2 6x

 + D D f  *D f*q28 + D D f  *D f *q46 + D D f  *D f  *q61
    1 2 4x  1         1 2 3x  1 x        1 2 2x  1 2x

    1               2
 - ---*D D f  *(D f) *q25 + D D f *D f  *q46 - 2*D D f *D f *D f*q21
    2   1 2 2x   1           1 2 x  1 3x          1 2 x  1 x  1

                                                       2
 + D D f*D f  *q28 - D D f*D f  *D f*q25 - D D f*(D f ) *q21
    1 2   1 4x        1 2   1 2x  1         1 2    1 x

    1             3
 - ---*D D f*(D f) *q19 - D f  *f *q29 - D f  *f  *q45 - D f  *D f*f *q24
    3   1 2    1           1 4x  x        1 3x  2x        1 2x  1   x

                    1        2
 - D f  *f  *q60 - ---*(D f ) *f *q22 - D f *D f*f  *q22 - D f *f  *q45
    1 2x  3x        2    1 x    x        1 x  1   2x        1 x  4x

        3           1       2
 + (D f) *f *q17 - ---*(D f) *f  *q24 - D f*f  *q29 + f  *q83
     1     x        2    1     3x        1   5x        7x

And now in machine readable form:

The system:

df(f(1),t)=d(1,d(2,f(1)))*p1 + df(f(1),x)*p2$

The symmetry:

df(f(1),s)=d(2,df(f(1),x,4))*d(1,d(2,f(1)))*q29 + d(2,df(f(1),x,4))*df(f(1),x)*
q28 + d(2,df(f(1),x,3))*d(1,d(2,df(f(1),x)))*q45 + d(2,df(f(1),x,3))*df(f(1),x,2
)*q46 + d(2,df(f(1),x,2))*d(2,f(1))*d(1,d(2,f(1)))*q25 + d(2,df(f(1),x,2))*d(2,f
(1))*df(f(1),x)*q24 + d(2,df(f(1),x,2))*d(1,d(2,df(f(1),x,2)))*q60 + d(2,df(f(1)
,x,2))*d(1,d(2,f(1)))*d(1,f(1))*q24 - d(2,df(f(1),x,2))*d(1,f(1))*df(f(1),x)*q25
 + d(2,df(f(1),x,2))*df(f(1),x,3)*q61 + d(2,df(f(1),x))**2*d(1,d(2,f(1)))*q21 + 
1/2*d(2,df(f(1),x))**2*df(f(1),x)*q22 + 2*d(2,df(f(1),x))*d(2,f(1))*d(1,d(2,df(f
(1),x)))*q21 + d(2,df(f(1),x))*d(2,f(1))*df(f(1),x,2)*q22 + d(2,df(f(1),x))*d(1,
d(2,df(f(1),x,3)))*q45 + d(2,df(f(1),x))*d(1,d(2,df(f(1),x)))*d(1,f(1))*q22 + d(
2,df(f(1),x))*d(1,d(2,f(1)))*d(1,df(f(1),x))*q22 - 2*d(2,df(f(1),x))*d(1,df(f(1)
,x))*df(f(1),x)*q21 - 2*d(2,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*q21 + d(2,df(f(1)
,x))*df(f(1),x,4)*q46 + d(2,f(1))**3*d(1,d(2,f(1)))*q17 + 1/3*d(2,f(1))**3*df(f(
1),x)*q19 + 1/2*d(2,f(1))**2*d(1,d(2,df(f(1),x,2)))*q25 + d(2,f(1))**2*d(1,d(2,f
(1)))*d(1,f(1))*q19 - 3*d(2,f(1))**2*d(1,f(1))*df(f(1),x)*q17 + 1/2*d(2,f(1))**2
*df(f(1),x,3)*q24 + d(2,f(1))*d(1,d(2,df(f(1),x,4)))*q29 + d(2,f(1))*d(1,d(2,df(
f(1),x,2)))*d(1,f(1))*q24 + d(2,f(1))*d(1,d(2,df(f(1),x)))*d(1,df(f(1),x))*q22 +
 d(2,f(1))*d(1,d(2,f(1)))*d(1,df(f(1),x,2))*q24 - 3*d(2,f(1))*d(1,d(2,f(1)))*d(1
,f(1))**2*q17 - d(2,f(1))*d(1,df(f(1),x,2))*df(f(1),x)*q25 - 2*d(2,f(1))*d(1,df(
f(1),x))*df(f(1),x,2)*q21 - d(2,f(1))*d(1,f(1))**2*df(f(1),x)*q19 - d(2,f(1))*d(
1,f(1))*df(f(1),x,3)*q25 + d(2,f(1))*df(f(1),x,5)*q28 + d(1,d(2,df(f(1),x,6)))*
q82 + d(1,d(2,df(f(1),x,4)))*d(1,f(1))*q28 + d(1,d(2,df(f(1),x,3)))*d(1,df(f(1),
x))*q46 + d(1,d(2,df(f(1),x,2)))*d(1,df(f(1),x,2))*q61 - 1/2*d(1,d(2,df(f(1),x,2
)))*d(1,f(1))**2*q25 + d(1,d(2,df(f(1),x)))*d(1,df(f(1),x,3))*q46 - 2*d(1,d(2,df
(f(1),x)))*d(1,df(f(1),x))*d(1,f(1))*q21 + d(1,d(2,f(1)))*d(1,df(f(1),x,4))*q28 
- d(1,d(2,f(1)))*d(1,df(f(1),x,2))*d(1,f(1))*q25 - d(1,d(2,f(1)))*d(1,df(f(1),x)
)**2*q21 - 1/3*d(1,d(2,f(1)))*d(1,f(1))**3*q19 - d(1,df(f(1),x,4))*df(f(1),x)*
q29 - d(1,df(f(1),x,3))*df(f(1),x,2)*q45 - d(1,df(f(1),x,2))*d(1,f(1))*df(f(1),x
)*q24 - d(1,df(f(1),x,2))*df(f(1),x,3)*q60 - 1/2*d(1,df(f(1),x))**2*df(f(1),x)*
q22 - d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*q22 - d(1,df(f(1),x))*df(f(1),x,4)*
q45 + d(1,f(1))**3*df(f(1),x)*q17 - 1/2*d(1,f(1))**2*df(f(1),x,3)*q24 - d(1,f(1)
)*df(f(1),x,5)*q29 + df(f(1),x,7)*q83$