Solution 2 to problem N2t2s14f3

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

Expressions

The solution is given through the following expressions:


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:

 q83, q82, q81, q80, q79, q78, q77, q76, q75, q74, q73, 
q72, q71, q70, q69, q68, q67, q66, q65, q64, q63, q62, 
q61, q60, q59, q58, q57, q56, q55, q54, q53, q52, q51, 
q50, q49, q48, q47, q46, q45, q44, q43, q42, q41, q40, 
q39, q38, q37, q36, q35, q34, q33, q32, q31, q30, q29, 
q28, q27, q26, q25, q24, q23, q22, q21, q20, q19, q18, 
q17, q16, q15, q14, q13, q12, q11, q10, q9, q8, q7, 
q6, q5, q4, q3, q2, q1, p2

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.

 
{g0139*q81 + g0140*q80 + g0141*q79 + g0142*q78 + g0143*q77 + g0144*q76

  + g0145*q75 + g0146*q74 + g0147*q73 + g0148*q72 + g0149*q71 + g0150*q70

  + g0151*q69 + g0152*q68 + g0153*q67 + g0154*q66 + g0155*q65 + g0156*q64

  + g0157*q63 + g0158*q62 + g0159*q61 + g0160*q60 + g0161*q59 + g0162*q58

  + g0163*q57 + g0164*q56 + g0165*q55 + g0166*q54 + g0167*q53 + g0168*q52

  + g0169*q51 + g0170*q50 + g0171*q49 + g0172*q48 + g0173*q47 + g0174*q46

  + g0175*q45 + g0176*q44 + g0177*q43 + g0178*q42 + g0179*q41 + g0180*q40

  + g0181*q39 + g0182*q38 + g0183*q37 + g0184*q36 + g0185*q35 + g0186*q34

  + g0187*q33 + g0188*q32 + g0189*q31 + g0190*q30 + g0191*q29 + g0192*q28

  + g0193*q27 + g0194*q26 + g0195*q25 + g0196*q24 + g0197*q23 + g0198*q22

  + g0199*q21 + g0200*q20 + g0201*q19 + g0202*q18 + g0203*q17 + g0204*q16

  + g0205*q15 + g0206*q14 + g0207*q13 + g0208*q12 + g0209*q11 + g0210*q10

  + g0211*q9 + g0212*q8 + g0213*q7 + g0214*q6 + g0215*q5 + g0216*q4 + g0217*q3

  + g0218*q2 + g0219*q1,

 g0003*q82 + g0004*q81 + g0005*q80 + g0006*q79 + g0007*q78 + g0008*q77

  + g0009*q76 + g0010*q74 + g0011*q73 + g0012*q71 + g0013*q70 + g0014*q69

  + g0015*q68 + g0016*q65 + g0017*q63 + g0018*q62 + g0019*q61 + g0020*q60

  + g0021*q59 + g0022*q58 + g0023*q57 + g0024*q56 + g0025*q55 + g0026*q53

  + g0027*q51 + g0028*q50 + g0029*q49 + g0030*q48 + g0031*q47 + g0032*q46

  + g0033*q45 + g0034*q44 + g0035*q42 + g0036*q40 + g0037*q39 + g0038*q37

  + g0039*q36 + g0040*q34 + g0041*q33 + g0042*q32 + g0043*q31 + g0044*q30

  + g0045*q29 + g0046*q28 + g0047*q27 + g0048*q26 + g0049*q25 + g0050*q24

  + g0051*q23 + g0052*q22 + g0053*q21 + g0054*q20 + g0055*q19 + g0056*q18

  + g0057*q17 + g0058*q16 + g0059*q15 + g0060*q14 + g0061*q12 + g0062*q11

  + g0063*q10 + g0064*q9 + g0065*q7 + g0066*q5 + g0067*q4 + g0068*q3 + g0069*q1,

 g0071*q82 + g0072*q81 + g0073*q80 + g0074*q79 + g0075*q78 + g0076*q77

  + g0077*q76 + g0078*q74 + g0079*q73 + g0080*q72 + g0081*q70 + g0082*q69

  + g0083*q67 + g0084*q66 + g0085*q64 + g0086*q63 + g0087*q61 + g0088*q60

  + g0089*q59 + g0090*q58 + g0091*q57 + g0092*q56 + g0093*q54 + g0094*q53

  + g0095*q52 + g0096*q50 + g0097*q49 + g0098*q48 + g0099*q47 + g0100*q46

  + g0101*q45 + g0102*q43 + g0103*q42 + g0104*q41 + g0105*q39 + g0106*q38

  + g0107*q37 + g0108*q35 + g0109*q34 + g0110*q33 + g0111*q31 + g0112*q30

  + g0113*q29 + g0114*q28 + g0115*q27 + g0116*q26 + g0117*q25 + g0118*q24

  + g0119*q23 + g0120*q22 + g0121*q21 + g0122*q20 + g0123*q19 + g0124*q18

  + g0125*q17 + g0126*q16 + g0127*q15 + g0128*q13 + g0129*q12 + g0130*q11

  + g0131*q10 + g0132*q8 + g0133*q7 + g0134*q6 + g0135*q5 + g0136*q4 + g0137*q2,

 p2}

Relevance for the application:

The equation:


f =f *p2
 t  x

The symmetry:

f =D f  *f*q1 + D f  *D D f*q29 + D f  *f *q44 + D f  *D f*f*q3
 s  2 5x         2 4x  1 2         2 4x  x        2 3x  2

 + D f  *D D f *q45 + D f  *D f*f*q4 + D f  *f  *q51 + D f  *D f *f*q9
    2 3x  1 2 x        2 3x  1          2 3x  2x        2 2x  2 x

 + D f  *D f*D D f*q25 + D f  *D f*f *q40 + D f  *D D f  *q60
    2 2x  2   1 2         2 2x  2   x        2 2x  1 2 2x

 + D f  *D D f*D f*q24 + D f  *D f *f*q10 + D f  *D f*f *q39 + D f  *f  *q65
    2 2x  1 2   1         2 2x  1 x          2 2x  1   x        2 2x  3x

         2                   2                    2
 + (D f ) *D D f*q21 + (D f ) *f *q36 + D f *(D f) *f*q14 + D f *D f*D D f *q49
     2 x    1 2          2 x    x        2 x   2             2 x  2   1 2 x

 + D f *D f*D f*f*q15 + D f *D f*f  *q55 + D f *D D f  *q70
    2 x  2   1           2 x  2   2x        2 x  1 2 3x

 + D f *D D f *D f*q50 + D f *D D f*D f *q22 + D f *D f  *f*q7
    2 x  1 2 x  1         2 x  1 2   1 x        2 x  1 2x

                                2
 + D f *D f *f *q37 + D f *(D f) *f*q16 + D f *D f*f  *q56 + D f *f  *q68
    2 x  1 x  x        2 x   1             2 x  1   2x        2 x  4x

        3                  3               2                    2
 + (D f) *D D f*q17 + (D f) *f *q32 + (D f) *D D f  *q57 + (D f) *D D f*D f*q19
     2     1 2          2     x         2     1 2 2x         2     1 2   1

        2                   2                   2
 + (D f) *D f *f*q11 + (D f) *D f*f *q34 + (D f) *f  *q62 + D f*D D f  *q73
     2     1 x           2     1   x         2     3x        2   1 2 4x

 + D f*D D f  *D f*q58 + D f*D D f *D f *q47 + D f*D D f*D f  *q27
    2   1 2 2x  1         2   1 2 x  1 x        2   1 2   1 2x

                  2
 + D f*D D f*(D f) *q18 - D f*D D f*f *f*q30 + D f*D f  *f*q5 + D f*D f  *f *q42
    2   1 2    1           2   1 2   x          2   1 3x         2   1 2x  x

                                                    2
 + D f*D f *D f*f*q12 + D f*D f *f  *q53 + D f*(D f) *f *q33 + D f*D f*f  *q63
    2   1 x  1           2   1 x  2x        2    1     x        2   1   3x

 + D f*f  *q71 + D D f  *q82 + D D f  *D f*q74 + D D f  *D f *q69
    2   5x        1 2 6x        1 2 4x  1         1 2 3x  1 x

                                                          2
 + D D f  *D D f*f*q78 + D D f  *D f  *q61 + D D f  *(D f) *q59
    1 2 2x  1 2           1 2 2x  1 2x        1 2 2x   1

 + D D f  *f *f*q77 - D D f *D D f*f *q81 + D D f *D f  *q46
    1 2 2x  x          1 2 x  1 2   x        1 2 x  1 3x

 + D D f *D f *D f*q48 - D D f *f  *f*q79 + D D f*D f  *q28
    1 2 x  1 x  1         1 2 x  2x          1 2   1 4x

                                     2                  3
 + D D f*D f  *D f*q26 + D D f*(D f ) *q23 + D D f*(D f) *q20
    1 2   1 2x  1         1 2    1 x          1 2    1

 - D D f*D f*f *f*q31 - D D f*f  *f*q76 + D D f*f  *f *q80 + D f  *f*q2
    1 2   1   x          1 2   3x          1 2   2x  x        1 5x

 + D f  *f *q43 + D f  *D f*f*q6 + D f  *f  *q52 + D f  *D f *f*q8
    1 4x  x        1 3x  1          1 3x  2x        1 2x  1 x

                                            2                    2
 + D f  *D f*f *q41 + D f  *f  *q66 + (D f ) *f *q38 + D f *(D f) *f*q13
    1 2x  1   x        1 2x  3x         1 x    x        1 x   1

                                          3               2
 + D f *D f*f  *q54 + D f *f  *q67 + (D f) *f *q35 + (D f) *f  *q64
    1 x  1   2x        1 x  4x         1     x         1     3x

 + D f*f  *q72 + f  *q83 + f  *f *f*q75
    1   5x        7x        3x  x

And now in machine readable form:

The system:

df(f(1),t)=df(f(1),x)*p2$

The symmetry:

df(f(1),s)=d(2,df(f(1),x,5))*f(1)*q1 + 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)*q44 + d(2,df(f(1),x,3))*d(2,f(1))*f(1)*q3 + d(2,df(f(
1),x,3))*d(1,d(2,df(f(1),x)))*q45 + d(2,df(f(1),x,3))*d(1,f(1))*f(1)*q4 + d(2,df
(f(1),x,3))*df(f(1),x,2)*q51 + d(2,df(f(1),x,2))*d(2,df(f(1),x))*f(1)*q9 + 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)*q40 + 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,df(f(1),x))*f(1)*q10 + d(2,df(f(
1),x,2))*d(1,f(1))*df(f(1),x)*q39 + d(2,df(f(1),x,2))*df(f(1),x,3)*q65 + d(2,df(
f(1),x))**2*d(1,d(2,f(1)))*q21 + d(2,df(f(1),x))**2*df(f(1),x)*q36 + d(2,df(f(1)
,x))*d(2,f(1))**2*f(1)*q14 + d(2,df(f(1),x))*d(2,f(1))*d(1,d(2,df(f(1),x)))*q49 
+ d(2,df(f(1),x))*d(2,f(1))*d(1,f(1))*f(1)*q15 + d(2,df(f(1),x))*d(2,f(1))*df(f(
1),x,2)*q55 + d(2,df(f(1),x))*d(1,d(2,df(f(1),x,3)))*q70 + d(2,df(f(1),x))*d(1,d
(2,df(f(1),x)))*d(1,f(1))*q50 + d(2,df(f(1),x))*d(1,d(2,f(1)))*d(1,df(f(1),x))*
q22 + d(2,df(f(1),x))*d(1,df(f(1),x,2))*f(1)*q7 + d(2,df(f(1),x))*d(1,df(f(1),x)
)*df(f(1),x)*q37 + d(2,df(f(1),x))*d(1,f(1))**2*f(1)*q16 + d(2,df(f(1),x))*d(1,f
(1))*df(f(1),x,2)*q56 + d(2,df(f(1),x))*df(f(1),x,4)*q68 + d(2,f(1))**3*d(1,d(2,
f(1)))*q17 + d(2,f(1))**3*df(f(1),x)*q32 + d(2,f(1))**2*d(1,d(2,df(f(1),x,2)))*
q57 + d(2,f(1))**2*d(1,d(2,f(1)))*d(1,f(1))*q19 + d(2,f(1))**2*d(1,df(f(1),x))*f
(1)*q11 + d(2,f(1))**2*d(1,f(1))*df(f(1),x)*q34 + d(2,f(1))**2*df(f(1),x,3)*q62 
+ d(2,f(1))*d(1,d(2,df(f(1),x,4)))*q73 + d(2,f(1))*d(1,d(2,df(f(1),x,2)))*d(1,f(
1))*q58 + d(2,f(1))*d(1,d(2,df(f(1),x)))*d(1,df(f(1),x))*q47 + d(2,f(1))*d(1,d(2
,f(1)))*d(1,df(f(1),x,2))*q27 + d(2,f(1))*d(1,d(2,f(1)))*d(1,f(1))**2*q18 - d(2,
f(1))*d(1,d(2,f(1)))*df(f(1),x)*f(1)*q30 + d(2,f(1))*d(1,df(f(1),x,3))*f(1)*q5 +
 d(2,f(1))*d(1,df(f(1),x,2))*df(f(1),x)*q42 + d(2,f(1))*d(1,df(f(1),x))*d(1,f(1)
)*f(1)*q12 + d(2,f(1))*d(1,df(f(1),x))*df(f(1),x,2)*q53 + d(2,f(1))*d(1,f(1))**2
*df(f(1),x)*q33 + d(2,f(1))*d(1,f(1))*df(f(1),x,3)*q63 + d(2,f(1))*df(f(1),x,5)*
q71 + d(1,d(2,df(f(1),x,6)))*q82 + d(1,d(2,df(f(1),x,4)))*d(1,f(1))*q74 + d(1,d(
2,df(f(1),x,3)))*d(1,df(f(1),x))*q69 + d(1,d(2,df(f(1),x,2)))*d(1,d(2,f(1)))*f(1
)*q78 + d(1,d(2,df(f(1),x,2)))*d(1,df(f(1),x,2))*q61 + d(1,d(2,df(f(1),x,2)))*d(
1,f(1))**2*q59 + d(1,d(2,df(f(1),x,2)))*df(f(1),x)*f(1)*q77 - d(1,d(2,df(f(1),x)
))*d(1,d(2,f(1)))*df(f(1),x)*q81 + d(1,d(2,df(f(1),x)))*d(1,df(f(1),x,3))*q46 + 
d(1,d(2,df(f(1),x)))*d(1,df(f(1),x))*d(1,f(1))*q48 - d(1,d(2,df(f(1),x)))*df(f(1
),x,2)*f(1)*q79 + 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))*q26 + d(1,d(2,f(1)))*d(1,df(f(1),x))**2*q23 + d(1,d(2,f(1)))
*d(1,f(1))**3*q20 - d(1,d(2,f(1)))*d(1,f(1))*df(f(1),x)*f(1)*q31 - d(1,d(2,f(1))
)*df(f(1),x,3)*f(1)*q76 + d(1,d(2,f(1)))*df(f(1),x,2)*df(f(1),x)*q80 + d(1,df(f(
1),x,5))*f(1)*q2 + d(1,df(f(1),x,4))*df(f(1),x)*q43 + d(1,df(f(1),x,3))*d(1,f(1)
)*f(1)*q6 + d(1,df(f(1),x,3))*df(f(1),x,2)*q52 + d(1,df(f(1),x,2))*d(1,df(f(1),x
))*f(1)*q8 + d(1,df(f(1),x,2))*d(1,f(1))*df(f(1),x)*q41 + d(1,df(f(1),x,2))*df(f
(1),x,3)*q66 + d(1,df(f(1),x))**2*df(f(1),x)*q38 + d(1,df(f(1),x))*d(1,f(1))**2*
f(1)*q13 + d(1,df(f(1),x))*d(1,f(1))*df(f(1),x,2)*q54 + d(1,df(f(1),x))*df(f(1),
x,4)*q67 + d(1,f(1))**3*df(f(1),x)*q35 + d(1,f(1))**2*df(f(1),x,3)*q64 + d(1,f(1
))*df(f(1),x,5)*q72 + df(f(1),x,7)*q83 + df(f(1),x,3)*df(f(1),x)*f(1)*q75$