Solution 24 to problem l1o35
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Expressions
The solution is given through the following expressions:
b75=0
b74=0
b73=0
b72=( - 25/729*a16**4*a7*b37)/a13**5
b71=( - 25/81*a16**3*a7*b37)/a13**4
b70=0
b69=( - 5/27*a16**3*a7*b37)/a13**4
b68=( - 5/9*a16**2*a7*b37)/a13**3
b67=( - 10/27*a16**2*a7*b37)/a13**3
b66=0
b65=0
b64=( - 5/9*a16**2*a7*b37)/a13**3
b63=0
b62=( - 10/81*a16**2*a7*b37)/a13**3
b61=( - 10/27*a16*a7*b37)/a13**2
b60=( - 5/9*a16*a7*b37)/a13**2
b59=0
b58=0
b57=0
b56=0
b55=(25/2187*a16**5*b37)/a13**5
b54=(5/81*a16**4*b37)/a13**4
b53=(170/729*a16**4*b37)/a13**4
b52=(10/27*a16**3*b37)/a13**3
b51=(10/9*a16**3*b37)/a13**3
b50=(65/81*a16**3*b37)/a13**3
b49=(130/243*a16**3*b37)/a13**3
b48=(10/9*a16**2*b37)/a13**2
b47=(10/3*a16**2*b37)/a13**2
b46=(5/3*a16**2*b37)/a13**2
b45=(20/9*a16**2*b37)/a13**2
b44=(110/81*a16**2*b37)/a13**2
b43=(55/81*a16**2*b37)/a13**2
b42=(5/3*a16*b37)/a13
b41=(10/3*a16*b37)/a13
b40=(10/3*a16*b37)/a13
b39=(5/3*a16*b37)/a13
b38=(10/27*a16*b37)/a13
b36=0
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
b28=(50/243*a16**3*a7*b37)/a13**4
b27=(25/81*a16**3*a7*b37)/a13**4
b26=(25/27*a16**2*a7*b37)/a13**3
b25=(25/27*a16**2*a7*b37)/a13**3
b24=(80/27*a16**2*a7*b37)/a13**3
b23=(5/9*a16**2*a7*b37)/a13**3
b22=(5/9*a16**2*a7*b37)/a13**3
b21=(5/3*a16*a7*b37)/a13**2
b20=(35/9*a16*a7*b37)/a13**2
b19=(10/3*a16*a7*b37)/a13**2
b18=(25/9*a16*a7*b37)/a13**2
b17=(65/27*a16*a7*b37)/a13**2
b16=(10/27*a16*a7*b37)/a13**2
b15=(10/9*a7*b37)/a13
b14=(25/9*a7*b37)/a13
b13=(5/3*a7*b37)/a13
b12=0
b11=( - 80/729*a16**4*b37)/a13**4
b10=( - 80/81*a16**3*b37)/a13**3
b9=( - 80/243*a16**3*b37)/a13**3
b8=( - 20/27*a16**2*b37)/a13**2
b7=( - 200/81*a16**2*b37)/a13**2
b6=( - 40/81*a16**2*b37)/a13**2
b5=( - 20/27*a16*b37)/a13
b4=( - 10/9*a16*b37)/a13
b3=( - 10/27*a16*b37)/a13
b2= - 1/9*b37
a22=( - 1/9*a16**2*a7)/a13**2
a21=( - 1/3*a16*a7)/a13
a20=0
a19=(1/27*a16**3)/a13**2
a18=(1/3*a16**2)/a13
a17=(1/3*a16**2)/a13
a15=a16
a14=1/3*a16
a12=0
a11=0
a10=(2/3*a16*a7)/a13
a9=(1/3*a16*a7)/a13
a8=a7
a6=0
a5=0
a4=0
a3=0
a2=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:
b37,a7,a16,a13
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\$
1
df(u,t)=(---*u *f(v,v)*a16*a7 + f(v ,v )*a13*a7 + f(v,v )*a13*a7
3 x x x 2x
2
+ ---*f(v,v )*a16*a7*u)/a13
3 x
1 2 2 1 2 3
df(v,t)=(---*u *a13 *a16*v + u *v *a13 *a16 + ---*u *a13*a16 *u*v + v *a13
3 2x x x 3 x 3x
2 1 2 2 1
+ v *a13 *a16*u + ---*v *a13*a16 *u - ---*f(v,v )*a13*a16*a7*v
2x 3 x 3 x
1 2 1 3 3 2
- ---*f(v,v)*a16 *a7*u*v + ----*a16 *u *v)/a13
9 27
1 4 10 3
df(u,s)=( - ---*u *a13 *b37 - ----*u *a13 *a16*b37*u
9 5x 27 4x
10 3 10 2
- ----*u *u *a13 *a16*b37 + ----*u *f(v,v)*a13 *a16*a7*b37
9 3x x 27 3x
40 2 2 2 20 2 3
- ----*u *a13 *a16 *b37*u - ----*u *a13 *a16*b37
81 3x 27 2x
200 2 2 65 2
- -----*u *u *a13 *a16 *b37*u + ----*u *f(v,v )*a13 *a16*a7*b37
81 2x x 27 2x x
5 2 80 3 3
+ ---*u *f(v,v)*a13*a16 *a7*b37*u - -----*u *a13*a16 *b37*u
9 2x 243 2x
20 3 2 2 5 2 2
- ----*u *a13 *a16 *b37 + ---*u *f(v,v)*a13*a16 *a7*b37
27 x 9 x
80 2 3 2 25 2
- ----*u *a13*a16 *b37*u + ----*u *f(v ,v )*a13 *a16*a7*b37
81 x 9 x x x
10 2 80 2
+ ----*u *f(v,v )*a13 *a16*a7*b37 + ----*u *f(v,v )*a13*a16 *a7*b37*u
3 x 2x 27 x x
25 3 2 80 4 4
+ ----*u *f(v,v)*a16 *a7*b37*u - -----*u *a16 *b37*u
81 x 729 x
5 3 25 3
+ ---*f(v ,v )*a13 *a7*b37 + ----*f(v ,v )*a13 *a7*b37
3 2x 2x 9 x 3x
35 2 25 2 2
+ ----*f(v ,v )*a13 *a16*a7*b37*u + ----*f(v ,v )*a13*a16 *a7*b37*u
9 x 2x 27 x x
10 3 5 2
+ ----*f(v,v )*a13 *a7*b37 + ---*f(v,v )*a13 *a16*a7*b37*u
9 4x 3 3x
25 2 2 50 3 3
+ ----*f(v,v )*a13*a16 *a7*b37*u + -----*f(v,v )*a16 *a7*b37*u )/
27 2x 243 x
4
a13
10 4 5 4
df(v,s)=(----*u *a13 *a16*b37*v + ---*u *v *a13 *a16*b37
27 4x 3 3x x
55 3 2 110 3 2
+ ----*u *a13 *a16 *b37*u*v + -----*u *u *a13 *a16 *b37*v
81 3x 81 2x x
10 4 20 3 2
+ ----*u *v *a13 *a16*b37 + ----*u *v *a13 *a16 *b37*u
3 2x 2x 9 2x x
10 2 2 130 2 3 2
- ----*u *f(v,v)*a13 *a16 *a7*b37*v + -----*u *a13 *a16 *b37*u *v
81 2x 243 2x
5 2 3 2 65 2 2 3
+ ---*u *v *a13 *a16 *b37 + ----*u *a13 *a16 *b37*u*v
3 x x 81 x
10 4 10 3 2
+ ----*u *v *a13 *a16*b37 + ----*u *v *a13 *a16 *b37*u
3 x 3x 3 x 2x
10 2 3 2 5 2 2
+ ----*u *v *a13 *a16 *b37*u - ---*u *f(v,v )*a13 *a16 *a7*b37*v
9 x x 9 x x
5 3 170 4 3
- ----*u *f(v,v)*a13*a16 *a7*b37*u*v + -----*u *a13*a16 *b37*u *v
27 x 729 x
5 5 4 10 3 2 2
+ v *a13 *b37 + ---*v *a13 *a16*b37*u + ----*v *a13 *a16 *b37*u
5x 3 4x 9 3x
10 2 3 3 5 4 4
+ ----*v *a13 *a16 *b37*u + ----*v *a13*a16 *b37*u
27 2x 81 x
5 3 10 2 2
- ---*f(v ,v )*a13 *a16*a7*b37*v - ----*f(v ,v )*a13 *a16 *a7*b37*u*v
9 x 2x 27 x x
10 3 5 2 2
- ----*f(v,v )*a13 *a16*a7*b37*v - ---*f(v,v )*a13 *a16 *a7*b37*u*v
27 3x 9 2x
25 3 2 25 4 3
- ----*f(v,v )*a13*a16 *a7*b37*u *v - -----*f(v,v)*a16 *a7*b37*u *v
81 x 729
25 5 5 5
+ ------*a16 *b37*u *v)/a13
2187
df(u,t)=(1/3*df(u,x)*f(v,v)*a16*a7 + f(df(v,x),df(v,x))*a13*a7 + f(v,df(v,x,2))*
a13*a7 + 2/3*f(v,df(v,x))*a16*a7*u)/a13$
df(v,t)=(1/3*df(u,x,2)*a13**2*a16*v + df(u,x)*df(v,x)*a13**2*a16 + 1/3*df(u,x)*
a13*a16**2*u*v + df(v,x,3)*a13**3 + df(v,x,2)*a13**2*a16*u + 1/3*df(v,x)*a13*a16
**2*u**2 - 1/3*f(v,df(v,x))*a13*a16*a7*v - 1/9*f(v,v)*a16**2*a7*u*v + 1/27*a16**
3*u**3*v)/a13**2$
df(u,s)=( - 1/9*df(u,x,5)*a13**4*b37 - 10/27*df(u,x,4)*a13**3*a16*b37*u - 10/9*
df(u,x,3)*df(u,x)*a13**3*a16*b37 + 10/27*df(u,x,3)*f(v,v)*a13**2*a16*a7*b37 - 40
/81*df(u,x,3)*a13**2*a16**2*b37*u**2 - 20/27*df(u,x,2)**2*a13**3*a16*b37 - 200/
81*df(u,x,2)*df(u,x)*a13**2*a16**2*b37*u + 65/27*df(u,x,2)*f(v,df(v,x))*a13**2*
a16*a7*b37 + 5/9*df(u,x,2)*f(v,v)*a13*a16**2*a7*b37*u - 80/243*df(u,x,2)*a13*a16
**3*b37*u**3 - 20/27*df(u,x)**3*a13**2*a16**2*b37 + 5/9*df(u,x)**2*f(v,v)*a13*
a16**2*a7*b37 - 80/81*df(u,x)**2*a13*a16**3*b37*u**2 + 25/9*df(u,x)*f(df(v,x),df
(v,x))*a13**2*a16*a7*b37 + 10/3*df(u,x)*f(v,df(v,x,2))*a13**2*a16*a7*b37 + 80/27
*df(u,x)*f(v,df(v,x))*a13*a16**2*a7*b37*u + 25/81*df(u,x)*f(v,v)*a16**3*a7*b37*u
**2 - 80/729*df(u,x)*a16**4*b37*u**4 + 5/3*f(df(v,x,2),df(v,x,2))*a13**3*a7*b37
+ 25/9*f(df(v,x),df(v,x,3))*a13**3*a7*b37 + 35/9*f(df(v,x),df(v,x,2))*a13**2*a16
*a7*b37*u + 25/27*f(df(v,x),df(v,x))*a13*a16**2*a7*b37*u**2 + 10/9*f(v,df(v,x,4)
)*a13**3*a7*b37 + 5/3*f(v,df(v,x,3))*a13**2*a16*a7*b37*u + 25/27*f(v,df(v,x,2))*
a13*a16**2*a7*b37*u**2 + 50/243*f(v,df(v,x))*a16**3*a7*b37*u**3)/a13**4$
df(v,s)=(10/27*df(u,x,4)*a13**4*a16*b37*v + 5/3*df(u,x,3)*df(v,x)*a13**4*a16*b37
+ 55/81*df(u,x,3)*a13**3*a16**2*b37*u*v + 110/81*df(u,x,2)*df(u,x)*a13**3*a16**
2*b37*v + 10/3*df(u,x,2)*df(v,x,2)*a13**4*a16*b37 + 20/9*df(u,x,2)*df(v,x)*a13**
3*a16**2*b37*u - 10/81*df(u,x,2)*f(v,v)*a13**2*a16**2*a7*b37*v + 130/243*df(u,x,
2)*a13**2*a16**3*b37*u**2*v + 5/3*df(u,x)**2*df(v,x)*a13**3*a16**2*b37 + 65/81*
df(u,x)**2*a13**2*a16**3*b37*u*v + 10/3*df(u,x)*df(v,x,3)*a13**4*a16*b37 + 10/3*
df(u,x)*df(v,x,2)*a13**3*a16**2*b37*u + 10/9*df(u,x)*df(v,x)*a13**2*a16**3*b37*u
**2 - 5/9*df(u,x)*f(v,df(v,x))*a13**2*a16**2*a7*b37*v - 5/27*df(u,x)*f(v,v)*a13*
a16**3*a7*b37*u*v + 170/729*df(u,x)*a13*a16**4*b37*u**3*v + df(v,x,5)*a13**5*b37
+ 5/3*df(v,x,4)*a13**4*a16*b37*u + 10/9*df(v,x,3)*a13**3*a16**2*b37*u**2 + 10/
27*df(v,x,2)*a13**2*a16**3*b37*u**3 + 5/81*df(v,x)*a13*a16**4*b37*u**4 - 5/9*f(
df(v,x),df(v,x,2))*a13**3*a16*a7*b37*v - 10/27*f(df(v,x),df(v,x))*a13**2*a16**2*
a7*b37*u*v - 10/27*f(v,df(v,x,3))*a13**3*a16*a7*b37*v - 5/9*f(v,df(v,x,2))*a13**
2*a16**2*a7*b37*u*v - 25/81*f(v,df(v,x))*a13*a16**3*a7*b37*u**2*v - 25/729*f(v,v
)*a16**4*a7*b37*u**3*v + 25/2187*a16**5*b37*u**5*v)/a13**5$