Old solution 4 to problem l112o13 before re-parametrization
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Back to problem l112o13
Expressions
The solution is given through the following expressions:
1 2 3 3
- ---*a0*a3*a4 *a7*b10 + ---*a3*a4 *a7*b1
8 4
b42=--------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
3 2 2 2 2
- ---*a0 *a3*a4*a7*b10 + 6*a0*a3*a4 *a7*b1 - 3*a3 *a4*a7 *b1
2
b40=---------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
3 2 2 3 3 2
---*a0 *a4 *b10 - ---*a0*a4 *b1 + 3*a3*a4 *a7*b1
4 2
b39=--------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 1 3 2 2 3
a0 *a4*a7 *b10 - ---*a0*a3*a7 *b10 - 6*a0*a4 *a7 *b1 + 3*a3*a4*a7 *b1
2
b38=-----------------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
2 2 2 2
- 2*a0 *a4*a7*b10 + a0*a3*a7 *b10 + 6*a0*a4 *a7*b1 - 12*a3*a4*a7 *b1
b37=-----------------------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 2
- 3*a0 *a4*a7*b10 + 6*a0*a4 *a7*b1 - 12*a3*a4*a7 *b1
b36=-------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
1 3 2 2 2 2
b35=( - ---*a0 *a4*a7*b10 + a0 *a3*a7 *b10 + 3*a0 *a4 *a7*b1
2
2 2 3 4 3
- 3*a0*a3*a4*a7 *b1 + 3*a3 *a7 *b1)/(a0 *a4 + a0 *a3*a7)
2 2 2
- a0 *a3*a7*b10 + 3*a0*a3*a4*a7*b1 - 3*a3 *a7 *b1
b34=----------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
3 3 1 2 2 2
b33=(---*a0 *a4*b10 + ---*a0 *a3*a7*b10 - 3*a0 *a4 *b1 + 6*a0*a3*a4*a7*b1
2 2
2 2 3 2
+ 3*a3 *a7 *b1)/(a0 *a4 + a0 *a3*a7)
2 2
3*a0 *a4*b10 - 6*a0*a4 *b1 + 12*a3*a4*a7*b1
b32=---------------------------------------------
2
a0 *a4 + a0*a3*a7
3 2 2
---*a0 *a4*b10 - 3*a0*a4 *b1 + 6*a3*a4*a7*b1
2
b31=----------------------------------------------
2
a0 *a4 + a0*a3*a7
3 3
- a0*a7 *b10 + 6*a4*a7 *b1
b30=-----------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 2 3
3*a0 *a7 *b10 - 6*a0*a4*a7 *b1 + 12*a3*a7 *b1
b29=-----------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 2 3
3*a0 *a7 *b10 - 12*a0*a4*a7 *b1 + 6*a3*a7 *b1
b28=-----------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2
- 3*a0 *a7*b10 + 6*a0*a4*a7*b1 - 12*a3*a7 *b1
b27=------------------------------------------------
2
a0 *a4 + a0*a3*a7
2 2
- 3*a0 *a7*b10 + 6*a0*a4*a7*b1 - 12*a3*a7 *b1
b26=------------------------------------------------
2
a0 *a4 + a0*a3*a7
2 2
- a0 *a7*b10 + 3*a0*a4*a7*b1 - 3*a3*a7 *b1
b25=---------------------------------------------
2
a0 *a4 + a0*a3*a7
2
a0 *b10 - 2*a0*a4*b1 + 4*a3*a7*b1
b24=-----------------------------------
a0*a4 + a3*a7
1 3 3 4
- ---*a0*a3*a4 *b10 + ---*a3*a4 *b1
8 4
b23=--------------------------------------
4 3
a0 *a4 + a0 *a3*a7
b22
1 2 3 1 2 3 4 3
- ---*a0 *a4 *b10 + ---*a0*a3*a4 *a7*b10 + ---*a0*a4 *b1 - 3*a3*a4 *a7*b1
4 2 2
=----------------------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
9 2 2 21 3 2 2
- ---*a0 *a3*a4 *b10 + ----*a0*a3*a4 *b1 - 3*a3 *a4 *a7*b1
4 2
b21=-------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
2 2 1 2 3 2 2
a0 *a4 *a7*b10 - ---*a0*a3*a4*a7 *b10 - 6*a0*a4 *a7*b1 + 3*a3*a4 *a7 *b1
2
b20=--------------------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
2 2 3 2
- 3*a0 *a4 *b10 + 3*a0*a3*a4*a7*b10 + 12*a0*a4 *b1 - 24*a3*a4 *a7*b1
b19=-----------------------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
3 2 2 3 2 2 2
---*a0 *a3*a4*a7*b10 + 3*a0 *a4 *b1 - 3*a0*a3*a4 *a7*b1 + 3*a3 *a4*a7 *b1
2
b18=---------------------------------------------------------------------------
4 3
a0 *a4 + a0 *a3*a7
2 2 2
- 3*a0 *a3*a4*b10 + 12*a0*a3*a4 *b1 - 6*a3 *a4*a7*b1
b17=-------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
5 2 2 2
- ---*a0 *a3*a4*b10 + 9*a0*a3*a4 *b1 - 6*a3 *a4*a7*b1
2
b16=--------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2
- a0*a3*a4*b10 + 6*a3*a4 *b1
b15=-------------------------------
2
a0 *a4 + a0*a3*a7
2 2 2
- a0*a4*a7 *b10 + 6*a4 *a7 *b1
b14=---------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 2 2
2*a0 *a4*a7*b10 - a0*a3*a7 *b10 - 12*a0*a4 *a7*b1 + 6*a3*a4*a7 *b1
b13=--------------------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2
2*a0*a4*a7*b10 - a3*a7 *b10 - 18*a4 *a7*b1
b12=--------------------------------------------
2
a0 *a4 + a0*a3*a7
2 2
- 2*a0 *a4*b10 + a0*a3*a7*b10 + 6*a0*a4 *b1 - 12*a3*a4*a7*b1
b11=---------------------------------------------------------------
2
a0 *a4 + a0*a3*a7
2 2 2 2 2
3*a0 *a3*a7*b10 + 6*a0 *a4 *b1 - 6*a0*a3*a4*a7*b1 + 6*a3 *a7 *b1
b9=------------------------------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2 2 2 2
a0 *a3*a7*b10 + 3*a0 *a4 *b1 + 3*a3 *a7 *b1
b8=---------------------------------------------
3 2
a0 *a4 + a0 *a3*a7
2 2
- a0 *a3*b10 + 3*a0*a3*a4*b1 - 3*a3 *a7*b1
b7=---------------------------------------------
2
a0 *a4 + a0*a3*a7
3*a3*b1
b6=---------
a0
b5=0
2
12*a7 *b1
b4=-----------
2
a0
- 6*a7*b1
b3=------------
a0
- 6*a7*b1
b2=------------
a0
1
---*a3*a7
2
a8=-----------
a0
1
---*a3*a4
2
a5=-----------
a0
a2=0
a1=a0 + 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:
b41, b10, b1, a6, a4, a3, a0, a7
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
df(u,t)=
2 1 2
u *a0 + u *a0*a6 + f(v,v )*a0*a3 + ---*f(v,v) *a3*a4 + f(v,v)*a0*a4*u
x x x 2
------------------------------------------------------------------------
a0
1
v *a0*a6 + ---*f(v,v)*a3*a7*v + a0*a7*u*v
x 2
df(v,t)=-------------------------------------------
a0
4 3 3
df(u,s)=(u *a0 *a4*b1 + u *a0 *a3*a7*b1 + u *f(v,v)*a0 *a3*a7*b10
3x 3x 2x
3 2 2 2
+ 3*u *f(v,v)*a0 *a4 *b1 + 3*u *f(v,v)*a0*a3 *a7 *b1
2x 2x
3 2 2 2 3
- 6*u *a0 *a4*a7*b1*u - 6*u *a0 *a3*a7 *b1*u - 6*u *a0 *a4*a7*b1
2x 2x x
2 2 2 3
- 6*u *a0 *a3*a7 *b1 + 3*u *f(v,v )*a0 *a3*a7*b10
x x x
3 2 2
+ 6*u *f(v,v )*a0 *a4 *b1 - 6*u *f(v,v )*a0 *a3*a4*a7*b1
x x x x
2 2 3 2 2
+ 6*u *f(v,v )*a0*a3 *a7 *b1 + ---*u *f(v,v) *a0 *a3*a4*a7*b10
x x 2 x
2 2 3 2 2
+ 3*u *f(v,v) *a0 *a4 *b1 - 3*u *f(v,v) *a0*a3*a4 *a7*b1
x x
2 2 2 3
+ 3*u *f(v,v) *a3 *a4*a7 *b1 + 2*u *f(v,v)*a0 *a4*a7*b10*u
x x
2 2 2 2
- u *f(v,v)*a0 *a3*a7 *b10*u - 18*u *f(v,v)*a0 *a4 *a7*b1*u
x x
2 2 2 3 2
+ 12*u *a0 *a4*a7 *b1*u + 12*u *a0*a3*a7 *b1*u
x x
3 2 2
+ 3*f(v ,v )*a0 *a3*a4*b1 + 3*f(v ,v )*a0 *a3 *a7*b1
x 2x x 2x
3 2 2
- f(v ,v )*f(v,v)*a0 *a3*a4*b10 + 6*f(v ,v )*f(v,v)*a0 *a3*a4 *b1
x x x x
4 3
+ f(v ,v )*a0 *a4*b10*u + f(v ,v )*a0 *a3*a7*b10*u
x x x x
4 3
- f(v,v )*a0 *a3*b10 + 3*f(v,v )*a0 *a3*a4*b1
3x 3x
2 2 5 3
- 3*f(v,v )*a0 *a3 *a7*b1 - ---*f(v,v )*f(v,v)*a0 *a3*a4*b10
3x 2 2x
2 2 2
+ 9*f(v,v )*f(v,v)*a0 *a3*a4 *b1 - 6*f(v,v )*f(v,v)*a0*a3 *a4*a7*b1
2x 2x
4 3
- 2*f(v,v )*a0 *a4*b10*u + f(v,v )*a0 *a3*a7*b10*u
2x 2x
3 2 2
+ 6*f(v,v )*a0 *a4 *b1*u - 12*f(v,v )*a0 *a3*a4*a7*b1*u
2x 2x
2 3 2 2 2
- 3*f(v,v ) *a0 *a3*a4*b10 + 12*f(v,v ) *a0 *a3*a4 *b1
x x
2 2 9 2 2 2
- 6*f(v,v ) *a0*a3 *a4*a7*b1 - ---*f(v,v )*f(v,v) *a0 *a3*a4 *b10
x 4 x
21 2 3 2 2 2
+ ----*f(v,v )*f(v,v) *a0*a3*a4 *b1 - 3*f(v,v )*f(v,v) *a3 *a4 *a7*b1
2 x x
3 2
- 3*f(v,v )*f(v,v)*a0 *a4 *b10*u
x
2
+ 3*f(v,v )*f(v,v)*a0 *a3*a4*a7*b10*u
x
2 3
+ 12*f(v,v )*f(v,v)*a0 *a4 *b1*u
x
2 3 2
- 24*f(v,v )*f(v,v)*a0*a3*a4 *a7*b1*u + 2*f(v,v )*a0 *a4*a7*b10*u
x x
2 2 2 2 2 2
- f(v,v )*a0 *a3*a7 *b10*u - 12*f(v,v )*a0 *a4 *a7*b1*u
x x
2 2 1 4 3
+ 6*f(v,v )*a0*a3*a4*a7 *b1*u - ---*f(v,v) *a0*a3*a4 *b10
x 8
3 4 4 1 3 2 3
+ ---*f(v,v) *a3*a4 *b1 - ---*f(v,v) *a0 *a4 *b10*u
4 4
1 3 2 3 3 4
+ ---*f(v,v) *a0*a3*a4 *a7*b10*u + ---*f(v,v) *a0*a4 *b1*u
2 2
3 3 2 2 2 2
- 3*f(v,v) *a3*a4 *a7*b1*u + f(v,v) *a0 *a4 *a7*b10*u
1 2 2 2 2 3 2
- ---*f(v,v) *a0*a3*a4*a7 *b10*u - 6*f(v,v) *a0*a4 *a7*b1*u
2
2 2 2 2 2 2 3
+ 3*f(v,v) *a3*a4 *a7 *b1*u - f(v,v)*a0 *a4*a7 *b10*u
2 2 3 4 3
+ 6*f(v,v)*a0*a4 *a7 *b1*u )/(a0 *a4 + a0 *a3*a7)
4 3 2 2
df(v,s)=( - u *a0 *a7*b10*v + 3*u *a0 *a4*a7*b1*v - 3*u *a0 *a3*a7 *b1*v
2x 2x 2x
4 3 2 2
- 3*u *v *a0 *a7*b10 + 6*u *v *a0 *a4*a7*b1 - 12*u *v *a0 *a3*a7 *b1
x x x x x x
1 3 2 2
- ---*u *f(v,v)*a0 *a4*a7*b10*v + u *f(v,v)*a0 *a3*a7 *b10*v
2 x x
2 2 2
+ 3*u *f(v,v)*a0 *a4 *a7*b1*v - 3*u *f(v,v)*a0*a3*a4*a7 *b1*v
x x
2 3 3 2
+ 3*u *f(v,v)*a3 *a7 *b1*v + 3*u *a0 *a7 *b10*u*v
x x
2 2 3 5
- 12*u *a0 *a4*a7 *b1*u*v + 6*u *a0*a3*a7 *b1*u*v + v *a0 *b10
x x 3x
4 3 3 4
- 2*v *a0 *a4*b1 + 4*v *a0 *a3*a7*b1 + ---*v *f(v,v)*a0 *a4*b10
3x 3x 2 2x
3 2 2
- 3*v *f(v,v)*a0 *a4 *b1 + 6*v *f(v,v)*a0 *a3*a4*a7*b1
2x 2x
4 3 2 2
- 3*v *a0 *a7*b10*u + 6*v *a0 *a4*a7*b1*u - 12*v *a0 *a3*a7 *b1*u
2x 2x 2x
4 3 2
+ 3*v *f(v,v )*a0 *a4*b10 - 6*v *f(v,v )*a0 *a4 *b1
x x x x
2 3 2 3 2
+ 12*v *f(v,v )*a0 *a3*a4*a7*b1 + ---*v *f(v,v) *a0 *a4 *b10
x x 4 x
3 2 2 3 2 2
- ---*v *f(v,v) *a0 *a4 *b1 + 3*v *f(v,v) *a0*a3*a4 *a7*b1
2 x x
3 2 2
- 3*v *f(v,v)*a0 *a4*a7*b10*u + 6*v *f(v,v)*a0 *a4 *a7*b1*u
x x
2 3 2 2
- 12*v *f(v,v)*a0*a3*a4*a7 *b1*u + 3*v *a0 *a7 *b10*u
x x
2 2 2 3 2
- 6*v *a0 *a4*a7 *b1*u + 12*v *a0*a3*a7 *b1*u
x x
3 4 1 3
+ ---*f(v ,v )*a0 *a4*b10*v + ---*f(v ,v )*a0 *a3*a7*b10*v
2 x x 2 x x
3 2 2
- 3*f(v ,v )*a0 *a4 *b1*v + 6*f(v ,v )*a0 *a3*a4*a7*b1*v
x x x x
2 2 3
+ 3*f(v ,v )*a0*a3 *a7 *b1*v - f(v,v )*a0 *a3*a7*b10*v
x x 2x
2 2 2
+ 3*f(v,v )*a0 *a3*a4*a7*b1*v - 3*f(v,v )*a0*a3 *a7 *b1*v
2x 2x
3 2
- ---*f(v,v )*f(v,v)*a0 *a3*a4*a7*b10*v
2 x
2
+ 6*f(v,v )*f(v,v)*a0*a3*a4 *a7*b1*v
x
2 2 3
- 3*f(v,v )*f(v,v)*a3 *a4*a7 *b1*v - 2*f(v,v )*a0 *a4*a7*b10*u*v
x x
2 2 2 2
+ f(v,v )*a0 *a3*a7 *b10*u*v + 6*f(v,v )*a0 *a4 *a7*b1*u*v
x x
2 1 3 2
- 12*f(v,v )*a0*a3*a4*a7 *b1*u*v - ---*f(v,v) *a0*a3*a4 *a7*b10*v
x 8
3 3 3 2 4
+ ---*f(v,v) *a3*a4 *a7*b1*v + f(v,v) *a0 *a4*b41*u*v
4
2 3 2 2 2
+ f(v,v) *a0 *a3*a7*b41*u*v + f(v,v)*a0 *a4*a7 *b10*u *v
1 3 2 2 2 2
- ---*f(v,v)*a0*a3*a7 *b10*u *v - 6*f(v,v)*a0*a4 *a7 *b1*u *v
2
3 2 2 3 3 3 3
+ 3*f(v,v)*a3*a4*a7 *b1*u *v - a0 *a7 *b10*u *v + 6*a0*a4*a7 *b1*u *v
4 3
)/(a0 *a4 + a0 *a3*a7)