Solution 1 to problem l1o24
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Expressions
The solution is given through the following expressions:
b39=0
1 3 5 3
- ----*a1*a8 *b8 - ----*a6*a8 *b8
12 24
b38=------------------------------------
3 7 4
a1*a6 + ---*a6
3
1 2 5 2
- ---*a1*a8 *b8 - ---*a6*a8 *b8
3 6
b37=----------------------------------
2 7 3
a1*a6 + ---*a6
3
b36=0
1 2 1 2
- ----*a1*a8 *b8 - ---*a6*a8 *b8
12 4
b35=-----------------------------------
2 7 3
a1*a6 + ---*a6
3
1 1
- ---*a1*a8*b8 - ---*a6*a8*b8
6 2
b34=--------------------------------
7 2
a1*a6 + ---*a6
3
1 1
- ---*a1*a8*b8 - ---*a6*a8*b8
6 3
b33=--------------------------------
7 2
a1*a6 + ---*a6
3
b32=0
b31=0
1 2 4 1 4 5 2 4
- ---*a1 *a8 *b8 - ---*a1*a6*a8 *b8 + ----*a6 *a8 *b8
6 3 24
b30=--------------------------------------------------------
4 7 5
a1*a4*a6 + ---*a4*a6
3
1 3
---*a8 *b8
3
b29=-----------------------
7 2
a1*a4*a6 + ---*a4*a6
3
2 3 3 3 2 3
- a1 *a8 *b8 - 2*a1*a6*a8 *b8 + ---*a6 *a8 *b8
2
b28=-------------------------------------------------
3 7 4
a1*a4*a6 + ---*a4*a6
3
2
a8 *b8
b27=-------------------
7
a1*a4 + ---*a4*a6
3
2
2*a8 *b8
b26=-------------------
7
a1*a4 + ---*a4*a6
3
1 2 2 2 2 2
- ---*a1 *a8 *b8 - a1*a6*a8 *b8 + a6 *a8 *b8
2
b25=-----------------------------------------------
2 7 3
a1*a4*a6 + ---*a4*a6
3
2 2 2 4 2 4 2 2
- ---*a1 *a8 *b8 - ---*a1*a6*a8 *b8 + ---*a6 *a8 *b8
3 3 3
b24=-------------------------------------------------------
2 7 3
a1*a4*a6 + ---*a4*a6
3
4
---*a6*a8*b8
3
b23=-------------------
7
a1*a4 + ---*a4*a6
3
2*a6*a8*b8
b22=-------------------
7
a1*a4 + ---*a4*a6
3
4
---*a6*a8*b8
3
b21=-------------------
7
a1*a4 + ---*a4*a6
3
1 2 1 1 2
- ---*a1 *a8*b8 - ---*a1*a6*a8*b8 + ---*a6 *a8*b8
6 3 2
b20=----------------------------------------------------
7 2
a1*a4*a6 + ---*a4*a6
3
2 2
---*a6 *b8
3
b19=-------------------
7
a1*a4 + ---*a4*a6
3
b18=0
b17=0
b16=0
1 2 5 2
---*a1*a8 *b8 + ---*a6*a8 *b8
3 6
b15=-------------------------------
2 7 3
a1*a6 + ---*a6
3
1 2 5 2
---*a1*a8 *b8 + ---*a6*a8 *b8
3 6
b14=-------------------------------
2 7 3
a1*a6 + ---*a6
3
2 5
---*a1*a8*b8 + ---*a6*a8*b8
3 3
b13=-----------------------------
7 2
a1*a6 + ---*a6
3
2 5
---*a1*a8*b8 + ---*a6*a8*b8
3 3
b12=-----------------------------
7 2
a1*a6 + ---*a6
3
8
a1*a8*b8 + ---*a6*a8*b8
3
b11=-------------------------
7 2
a1*a6 + ---*a6
3
1 1
---*a1*a8*b8 + ---*a6*a8*b8
6 2
b10=-----------------------------
7 2
a1*a6 + ---*a6
3
1
---*a1*b8 + a6*b8
3
b9=-------------------
7
a1 + ---*a6
3
b7=0
4 2 3 8 3 4 2 3
---*a1 *a8 *b8 + ---*a1*a6*a8 *b8 - ---*a6 *a8 *b8
3 3 3
b6=----------------------------------------------------
3 7 4
a1*a4*a6 + ---*a4*a6
3
2 2 2 2 2
4*a1 *a8 *b8 + 8*a1*a6*a8 *b8 - 4*a6 *a8 *b8
b5=----------------------------------------------
2 7 3
a1*a4*a6 + ---*a4*a6
3
2 2 2 2 2
2*a1 *a8 *b8 + 4*a1*a6*a8 *b8 - 2*a6 *a8 *b8
b4=----------------------------------------------
2 7 3
a1*a4*a6 + ---*a4*a6
3
10 2 20 10 2
----*a1 *a8*b8 + ----*a1*a6*a8*b8 - ----*a6 *a8*b8
3 3 3
b3=----------------------------------------------------
7 2
a1*a4*a6 + ---*a4*a6
3
4 2 8 4 2
---*a1 *a8*b8 + ---*a1*a6*a8*b8 - ---*a6 *a8*b8
3 3 3
b2=-------------------------------------------------
7 2
a1*a4*a6 + ---*a4*a6
3
1 2 2 1 2
---*a1 *b8 + ---*a1*a6*b8 - ---*a6 *b8
3 3 3
b1=----------------------------------------
7
a1*a4 + ---*a4*a6
3
1
- ---*a4*a8
4
a10=--------------
a6
1 2 1 2
- ---*a1*a8 + ---*a6*a8
2 4
a9=----------------------------
2
a6
1 1
- ---*a1*a8 + ---*a6*a8
2 2
a7=--------------------------
a6
a5=0
a3=0
2*a1*a8
a2=---------
a6
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:
b8,a4,a8,a6,a1
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
u *a1*a6 + 2*u *a1*a8*u + f(v,v )*a4*a6
2x x x
df(u,t)=------------------------------------------
a6
1 2 3 2 4 1 5
df(u,s)=(---*u *a1 *a6 *b8 + ---*u *a1*a6 *b8 - ---*u *a6 *b8
3 4x 3 4x 3 4x
4 2 2 8 3
+ ---*u *a1 *a6 *a8*b8*u + ---*u *a1*a6 *a8*b8*u
3 3x 3 3x
4 4 10 2 2
- ---*u *a6 *a8*b8*u + ----*u *u *a1 *a6 *a8*b8
3 3x 3 2x x
20 3 10 4
+ ----*u *u *a1*a6 *a8*b8 - ----*u *u *a6 *a8*b8
3 2x x 3 2x x
1 2 1 3
+ ---*u *f(v,v)*a1*a4*a6 *a8*b8 + ---*u *f(v,v)*a4*a6 *a8*b8
6 2x 2 2x
2 2 2 2 2 2
+ 2*u *a1 *a6*a8 *b8*u + 4*u *a1*a6 *a8 *b8*u
2x 2x
3 2 2 2 2 2 2 2 2
- 2*u *a6 *a8 *b8*u + 4*u *a1 *a6*a8 *b8*u + 8*u *a1*a6 *a8 *b8*u
2x x x
2 3 2 2
- 4*u *a6 *a8 *b8*u + u *f(v,v )*a1*a4*a6 *a8*b8
x x x
8 3 1 2
+ ---*u *f(v,v )*a4*a6 *a8*b8 + ---*u *f(v,v)*a1*a4*a6*a8 *b8*u
3 x x 3 x
5 2 2 4 2 3 3
+ ---*u *f(v,v)*a4*a6 *a8 *b8*u + ---*u *a1 *a8 *b8*u
6 x 3 x
8 3 3 4 2 3 3
+ ---*u *a1*a6*a8 *b8*u - ---*u *a6 *a8 *b8*u
3 x 3 x
3 7 4
+ f(v ,v )*a1*a4*a6 *b8 + ---*f(v ,v )*a4*a6 *b8
x 2x 3 x 2x
2 2 5 3
+ ---*f(v ,v )*a1*a4*a6 *a8*b8*u + ---*f(v ,v )*a4*a6 *a8*b8*u
3 x x 3 x x
1 3 4
+ ---*f(v,v )*a1*a4*a6 *b8 + f(v,v )*a4*a6 *b8
3 3x 3x
2 2 5 3
+ ---*f(v,v )*a1*a4*a6 *a8*b8*u + ---*f(v,v )*a4*a6 *a8*b8*u
3 2x 3 2x
1 2 2 5 2 2 2
+ ---*f(v,v )*a1*a4*a6*a8 *b8*u + ---*f(v,v )*a4*a6 *a8 *b8*u )/(
3 x 6 x
3 7 4
a1*a4*a6 + ---*a4*a6 )
3
1 1 2 3 2
df(v,t)=( - ---*u *a1*a6*a8*v + ---*u *a6 *a8*v + v *a6 + v *a6 *a8*u
2 x 2 x 2x x
1 1 2 2 1 2 2 2
- ---*f(v,v)*a4*a6*a8*v - ---*a1*a8 *u *v + ---*a6*a8 *u *v)/a6
4 2 4
1 2 3 1 4
df(v,s)=( - ---*u *a1 *a6 *a8*b8*v - ---*u *a1*a6 *a8*b8*v
6 3x 3 3x
1 5 4 5
+ ---*u *a6 *a8*b8*v + ---*u *v *a6 *a8*b8
2 3x 3 2x x
2 2 2 2 4 3 2
- ---*u *a1 *a6 *a8 *b8*u*v - ---*u *a1*a6 *a8 *b8*u*v
3 2x 3 2x
4 4 2 1 2 2 2 2
+ ---*u *a6 *a8 *b8*u*v - ---*u *a1 *a6 *a8 *b8*v
3 2x 2 x
2 3 2 2 4 2 5
- u *a1*a6 *a8 *b8*v + u *a6 *a8 *b8*v + 2*u *v *a6 *a8*b8
x x x 2x
4 2 1 2 2
+ 2*u *v *a6 *a8 *b8*u - ----*u *f(v,v)*a1*a4*a6 *a8 *b8*v
x x 12 x
1 3 2 2 3 2
- ---*u *f(v,v)*a4*a6 *a8 *b8*v - u *a1 *a6*a8 *b8*u *v
4 x x
2 3 2 3 3 3 2 2 6
- 2*u *a1*a6 *a8 *b8*u *v + ---*u *a6 *a8 *b8*u *v + ---*v *a6 *b8
x 2 x 3 4x
4 5 4 2 2 1 3 3 3
+ ---*v *a6 *a8*b8*u + v *a6 *a8 *b8*u + ---*v *a6 *a8 *b8*u
3 3x 2x 3 x
1 3 1 4
- ---*f(v ,v )*a1*a4*a6 *a8*b8*v - ---*f(v ,v )*a4*a6 *a8*b8*v
6 x x 3 x x
1 3 1 4
- ---*f(v,v )*a1*a4*a6 *a8*b8*v - ---*f(v,v )*a4*a6 *a8*b8*v
6 2x 2 2x
1 2 2 5 3 2
- ---*f(v,v )*a1*a4*a6 *a8 *b8*u*v - ---*f(v,v )*a4*a6 *a8 *b8*u*v
3 x 6 x
1 3 2 5 2 3 2
- ----*f(v,v)*a1*a4*a6*a8 *b8*u *v - ----*f(v,v)*a4*a6 *a8 *b8*u *v
12 24
1 2 4 4 1 4 4 5 2 4 4
- ---*a1 *a8 *b8*u *v - ---*a1*a6*a8 *b8*u *v + ----*a6 *a8 *b8*u *v)/
6 3 24
4 7 5
(a1*a4*a6 + ---*a4*a6 )
3
df(u,t)=(df(u,x,2)*a1*a6 + 2*df(u,x)*a1*a8*u + f(v,df(v,x))*a4*a6)/a6$
df(u,s)=(1/3*df(u,x,4)*a1**2*a6**3*b8 + 2/3*df(u,x,4)*a1*a6**4*b8 - 1/3*df(u,x,4
)*a6**5*b8 + 4/3*df(u,x,3)*a1**2*a6**2*a8*b8*u + 8/3*df(u,x,3)*a1*a6**3*a8*b8*u
- 4/3*df(u,x,3)*a6**4*a8*b8*u + 10/3*df(u,x,2)*df(u,x)*a1**2*a6**2*a8*b8 + 20/3*
df(u,x,2)*df(u,x)*a1*a6**3*a8*b8 - 10/3*df(u,x,2)*df(u,x)*a6**4*a8*b8 + 1/6*df(u
,x,2)*f(v,v)*a1*a4*a6**2*a8*b8 + 1/2*df(u,x,2)*f(v,v)*a4*a6**3*a8*b8 + 2*df(u,x,
2)*a1**2*a6*a8**2*b8*u**2 + 4*df(u,x,2)*a1*a6**2*a8**2*b8*u**2 - 2*df(u,x,2)*a6
**3*a8**2*b8*u**2 + 4*df(u,x)**2*a1**2*a6*a8**2*b8*u + 8*df(u,x)**2*a1*a6**2*a8
**2*b8*u - 4*df(u,x)**2*a6**3*a8**2*b8*u + df(u,x)*f(v,df(v,x))*a1*a4*a6**2*a8*
b8 + 8/3*df(u,x)*f(v,df(v,x))*a4*a6**3*a8*b8 + 1/3*df(u,x)*f(v,v)*a1*a4*a6*a8**2
*b8*u + 5/6*df(u,x)*f(v,v)*a4*a6**2*a8**2*b8*u + 4/3*df(u,x)*a1**2*a8**3*b8*u**3
+ 8/3*df(u,x)*a1*a6*a8**3*b8*u**3 - 4/3*df(u,x)*a6**2*a8**3*b8*u**3 + f(df(v,x)
,df(v,x,2))*a1*a4*a6**3*b8 + 7/3*f(df(v,x),df(v,x,2))*a4*a6**4*b8 + 2/3*f(df(v,x
),df(v,x))*a1*a4*a6**2*a8*b8*u + 5/3*f(df(v,x),df(v,x))*a4*a6**3*a8*b8*u + 1/3*f
(v,df(v,x,3))*a1*a4*a6**3*b8 + f(v,df(v,x,3))*a4*a6**4*b8 + 2/3*f(v,df(v,x,2))*
a1*a4*a6**2*a8*b8*u + 5/3*f(v,df(v,x,2))*a4*a6**3*a8*b8*u + 1/3*f(v,df(v,x))*a1*
a4*a6*a8**2*b8*u**2 + 5/6*f(v,df(v,x))*a4*a6**2*a8**2*b8*u**2)/(a1*a4*a6**3 + 7/
3*a4*a6**4)$
df(v,t)=( - 1/2*df(u,x)*a1*a6*a8*v + 1/2*df(u,x)*a6**2*a8*v + df(v,x,2)*a6**3 +
df(v,x)*a6**2*a8*u - 1/4*f(v,v)*a4*a6*a8*v - 1/2*a1*a8**2*u**2*v + 1/4*a6*a8**2*
u**2*v)/a6**2$
df(v,s)=( - 1/6*df(u,x,3)*a1**2*a6**3*a8*b8*v - 1/3*df(u,x,3)*a1*a6**4*a8*b8*v +
1/2*df(u,x,3)*a6**5*a8*b8*v + 4/3*df(u,x,2)*df(v,x)*a6**5*a8*b8 - 2/3*df(u,x,2)
*a1**2*a6**2*a8**2*b8*u*v - 4/3*df(u,x,2)*a1*a6**3*a8**2*b8*u*v + 4/3*df(u,x,2)*
a6**4*a8**2*b8*u*v - 1/2*df(u,x)**2*a1**2*a6**2*a8**2*b8*v - df(u,x)**2*a1*a6**3
*a8**2*b8*v + df(u,x)**2*a6**4*a8**2*b8*v + 2*df(u,x)*df(v,x,2)*a6**5*a8*b8 + 2*
df(u,x)*df(v,x)*a6**4*a8**2*b8*u - 1/12*df(u,x)*f(v,v)*a1*a4*a6**2*a8**2*b8*v -
1/4*df(u,x)*f(v,v)*a4*a6**3*a8**2*b8*v - df(u,x)*a1**2*a6*a8**3*b8*u**2*v - 2*df
(u,x)*a1*a6**2*a8**3*b8*u**2*v + 3/2*df(u,x)*a6**3*a8**3*b8*u**2*v + 2/3*df(v,x,
4)*a6**6*b8 + 4/3*df(v,x,3)*a6**5*a8*b8*u + df(v,x,2)*a6**4*a8**2*b8*u**2 + 1/3*
df(v,x)*a6**3*a8**3*b8*u**3 - 1/6*f(df(v,x),df(v,x))*a1*a4*a6**3*a8*b8*v - 1/3*f
(df(v,x),df(v,x))*a4*a6**4*a8*b8*v - 1/6*f(v,df(v,x,2))*a1*a4*a6**3*a8*b8*v - 1/
2*f(v,df(v,x,2))*a4*a6**4*a8*b8*v - 1/3*f(v,df(v,x))*a1*a4*a6**2*a8**2*b8*u*v -
5/6*f(v,df(v,x))*a4*a6**3*a8**2*b8*u*v - 1/12*f(v,v)*a1*a4*a6*a8**3*b8*u**2*v -
5/24*f(v,v)*a4*a6**2*a8**3*b8*u**2*v - 1/6*a1**2*a8**4*b8*u**4*v - 1/3*a1*a6*a8
**4*b8*u**4*v + 5/24*a6**2*a8**4*b8*u**4*v)/(a1*a4*a6**4 + 7/3*a4*a6**5)$