Solution 23 to problem l112o23
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Expressions
The solution is given through the following expressions:
1 2 2 1 2 2
---*a14 *a4 *b32 + ---*a14 *a4 *b33
4 4
b42=-------------------------------------
4
a1
2 2
a14 *a4*b32 + a14 *a4*b33
b41=---------------------------
3
a1
3 4 1 2 5 2 2
b40=( - ---*a1 *b19 - ---*a1 *a14*a4*b31 - ---*a1 *a14*a4*b32 - 2*a1 *a14*a4*b33
2 2 2
11 2 2 4
+ ----*a14 *a4 *b1)/a1
2
1 2 1 2 1 2 2
---*a1 *a14*a4*b31 + ---*a1 *a14*a4*b32 - ---*a14 *a4 *b1
2 2 4
b39=-----------------------------------------------------------
4
a1
2 2
a14 *b32 + a14 *b33
b38=---------------------
2
a1
4 2 2 2 2
- a1 *b19 - a1 *a14*a4*b31 - a1 *a14*a4*b32 + 3*a14 *a4 *b1
b37=--------------------------------------------------------------
3
a1 *a4
a14*b31 + a14*b32
b36=-------------------
a1
4 2 2 2 2
- a1 *b19 - 2*a1 *a14*a4*b32 - 2*a1 *a14*a4*b33 + 3*a14 *a4 *b1
b35=------------------------------------------------------------------
3
a1 *a4
4 2 2 2
b34=( - a1 *b19 - a1 *a14*a4*b31 - 2*a1 *a14*a4*b32 - 2*a1 *a14*a4*b33
2 2 2
+ 3*a14 *a4 *b1)/(a1 *a14*a4)
b30=0
2
a14 *b1
b29=---------
2
a1
2
- 2*a14 *b1
b28=--------------
2
a1
b27=0
a14*b1
b26=--------
a1
- a14*b1
b25=-----------
a1
b24=0
1 2 3 1 2 3
- ---*a14 *a4 *b32 - ---*a14 *a4 *b33
4 4
b23=----------------------------------------
5
a1
2 2 2 2
- a14 *a4 *b32 - a14 *a4 *b33
b22=--------------------------------
4
a1
3 4 2 2 2 2 9 2 3
---*a1 *a4*b19 + 2*a1 *a14*a4 *b32 + 2*a1 *a14*a4 *b33 - ---*a14 *a4 *b1
2 2
b21=--------------------------------------------------------------------------
5
a1
2 2
- a14 *a4*b32 - a14 *a4*b33
b20=------------------------------
3
a1
4 2 2 9 2 2
a1 *b19 + 2*a1 *a14*a4*b32 + 2*a1 *a14*a4*b33 - ---*a14 *a4 *b1
4
b18=-----------------------------------------------------------------
4
a1
2 2
- a1 *a4*b32 + 3*a14*a4 *b1
b17=------------------------------
3
a1
4 2 2 3 2 2
a1 *b19 + 2*a1 *a14*a4*b32 + 2*a1 *a14*a4*b33 - ---*a14 *a4 *b1
2
b16=-----------------------------------------------------------------
3
a1 *a14
2 3 2
- a1 *a4*b33 + ---*a14*a4 *b1
2
b15=--------------------------------
3
a1
b14=0
2
2*a14 *a4*b1
b13=--------------
3
a1
2
5*a14 *a4*b1
b12=--------------
3
a1
3*a14*a4*b1
b11=-------------
2
a1
3*a14*a4*b1
b10=-------------
2
a1
5*a14*a4*b1
b9=-------------
2
a1
5
---*a14*a4*b1
2
b8=---------------
2
a1
a4*b1
b7=-------
a1
3*a4*b1
b6=---------
a1
b5=0
2
3*a14 *b1
b4=-----------
2
a1
3*a14*b1
b3=----------
a1
3*a14*b1
b2=----------
a1
a19=0
a18=0
- a14*a4
a17=-----------
a1
1
---*a14*a4
2
a16=------------
a1
a15=0
a13= - a14
a12=0
a11=0
a10=0
3 2
---*a14*a4
2
a9=-------------
2
a1
a8=0
a14*a4
a7=--------
a1
2*a14*a4
a6=----------
a1
a5=a4
a3=0
a2=2*a14
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:
b33,b32,b19,b31,b1,a4,a14,a1
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
3 2 2
df(u,t)=(u *a1 + 2*u *f(v,v)*a1*a14*a4 + 2*u *a1 *a14*u + f(v ,v )*a1 *a4
2x x x x x
2 3 2
+ f(v,v )*a1 *a4 + ---*f(v,v )*f(v,v)*a14*a4 + f(v,v )*a1*a14*a4*u)/
2x 2 x x
2
a1
1
- u *a1*a14*v + ---*v *f(v,v)*a14*a4 + v *a1*a14*u - f(v,v )*a14*a4*v
x 2 x x x
df(v,t)=------------------------------------------------------------------------
a1
5 5 3 2 4 2
df(u,s)=(u *a1 *a14*b1 + ---*u *f(v,v)*a1 *a14 *a4*b1 + 3*u *a1 *a14 *b1*u
3x 2 2x 2x
2 4 2 3 2
+ 3*u *a1 *a14 *b1 + 5*u *f(v,v )*a1 *a14 *a4*b1
x x x
2 5 2 3 2
+ u *f(v,v) *a1 *a14*b19 + 2*u *f(v,v) *a1 *a14 *a4*b32
x x
2 3 2 9 2 3 2
+ 2*u *f(v,v) *a1 *a14 *a4*b33 - ---*u *f(v,v) *a1*a14 *a4 *b1
x 4 x
2 3 3 3 2
+ 5*u *f(v,v)*a1 *a14 *a4*b1*u + 3*u *a1 *a14 *b1*u
x x
4 4
+ 3*f(v ,v )*a1 *a14*a4*b1 - f(v ,v )*f(v,v)*a1 *a14*a4*b33
x 2x x x
3 2 2 2 3 2
+ ---*f(v ,v )*f(v,v)*a1 *a14 *a4 *b1 + 3*f(v ,v )*a1 *a14 *a4*b1*u
2 x x x x
4 6
+ f(v,v )*a1 *a14*a4*b1 + f(v,v )*f(v,v)*a1 *b19
3x 2x
4 4
+ 2*f(v,v )*f(v,v)*a1 *a14*a4*b32 + 2*f(v,v )*f(v,v)*a1 *a14*a4*b33
2x 2x
3 2 2 2 3 2
- ---*f(v,v )*f(v,v)*a1 *a14 *a4 *b1 + 3*f(v,v )*a1 *a14 *a4*b1*u
2 2x 2x
2 4 2 2 2 2
- f(v,v ) *a1 *a14*a4*b32 + 3*f(v,v ) *a1 *a14 *a4 *b1
x x
3 2 4
+ ---*f(v,v )*f(v,v) *a1 *a14*a4*b19
2 x
2 2 2 2
+ 2*f(v,v )*f(v,v) *a1 *a14 *a4 *b32
x
2 2 2 2 9 2 3 3
+ 2*f(v,v )*f(v,v) *a1 *a14 *a4 *b33 - ---*f(v,v )*f(v,v) *a14 *a4 *b1
x 2 x
5 2 3 2
+ f(v,v )*f(v,v)*a1 *a14*b19*u + 2*f(v,v )*a1 *a14 *a4*b1*u
x x
1 4 3 3 1 4 3 3
- ---*f(v,v) *a14 *a4 *b32 - ---*f(v,v) *a14 *a4 *b33
4 4
3 3 2 3 3 2
- f(v,v) *a1*a14 *a4 *b32*u - f(v,v) *a1*a14 *a4 *b33*u
2 2 3 2 2 2 3 2 5
- f(v,v) *a1 *a14 *a4*b32*u - f(v,v) *a1 *a14 *a4*b33*u )/(a1 *a14)
3 2 3 2
df(v,s)=( - u *a1 *a14 *a4*b1*v + u *v *a1 *a14 *a4*b1
2x x x
5 3 2
- u *f(v,v)*a1 *a14*b19*v - 2*u *f(v,v)*a1 *a14 *a4*b32*v
x x
3 2 3 2
- 2*u *f(v,v)*a1 *a14 *a4*b33*v + 3*u *f(v,v)*a1*a14 *a4 *b1*v
x x
2 3 4
- 2*u *a1 *a14 *a4*b1*u*v + v *f(v,v)*a1 *a14*a4*b31
x 2x
4 1 2 2 2 2
+ v *f(v,v )*a1 *a14*a4*b32 + ---*v *f(v,v) *a1 *a14 *a4 *b31
x x 2 x
1 2 2 2 2 1 2 3 3
+ ---*v *f(v,v) *a1 *a14 *a4 *b32 - ---*v *f(v,v) *a14 *a4 *b1
2 x 4 x
3 2 3 2
+ v *f(v,v)*a1 *a14 *a4*b31*u + v *f(v,v)*a1 *a14 *a4*b32*u
x x
2 3 2 4
+ v *a1 *a14 *a4*b1*u + f(v ,v )*a1 *a14*a4*b33*v
x x x
6 4
- f(v,v )*a1 *b19*v - f(v,v )*a1 *a14*a4*b31*v
2x 2x
4 4
- 2*f(v,v )*a1 *a14*a4*b32*v - 2*f(v,v )*a1 *a14*a4*b33*v
2x 2x
2 2 2 3 4
+ 3*f(v,v )*a1 *a14 *a4 *b1*v - ---*f(v,v )*f(v,v)*a1 *a14*a4*b19*v
2x 2 x
1 2 2 2
- ---*f(v,v )*f(v,v)*a1 *a14 *a4 *b31*v
2 x
5 2 2 2
- ---*f(v,v )*f(v,v)*a1 *a14 *a4 *b32*v
2 x
2 2 2
- 2*f(v,v )*f(v,v)*a1 *a14 *a4 *b33*v
x
11 3 3 5
+ ----*f(v,v )*f(v,v)*a14 *a4 *b1*v - f(v,v )*a1 *a14*b19*u*v
2 x x
3 2 3 2
- f(v,v )*a1 *a14 *a4*b31*u*v - f(v,v )*a1 *a14 *a4*b32*u*v
x x
3 2 1 3 3 3
+ 3*f(v,v )*a1*a14 *a4 *b1*u*v + ---*f(v,v) *a14 *a4 *b32*v
x 4
1 3 3 3 2 3 2
+ ---*f(v,v) *a14 *a4 *b33*v + f(v,v) *a1*a14 *a4 *b32*u*v
4
2 3 2 2 3 2
+ f(v,v) *a1*a14 *a4 *b33*u*v + f(v,v)*a1 *a14 *a4*b32*u *v
2 3 2 4
+ f(v,v)*a1 *a14 *a4*b33*u *v)/(a1 *a14*a4)