Solution 25 to problem l1o35
Expressions |
Parameters |
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Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
b73=0
b72=0
25 2 20 2
----*a12*a15 *a21*b36 + ----*a15 *a1*a21*b36
81 81
b71=----------------------------------------------
4
a12
25 20
----*a12*a15*a21*b36 + ----*a15*a1*a21*b36
9 9
b70=--------------------------------------------
3
a12
b69=0
5 10
---*a12*a15*a21*b36 + ----*a15*a1*a21*b36
3 9
b68=-------------------------------------------
3
a12
10
5*a12*a21*b36 + ----*a1*a21*b36
3
b67=---------------------------------
2
a12
10 10
----*a12*a21*b36 + ----*a1*a21*b36
3 3
b66=------------------------------------
2
a12
b65=0
b64=0
10
5*a12*a21*b36 + ----*a1*a21*b36
3
b63=---------------------------------
2
a12
b62=0
10 5
----*a12*a21*b36 + ---*a1*a21*b36
9 9
b61=-----------------------------------
2
a12
10 5
----*a12*a21*b36 + ---*a1*a21*b36
3 3
b60=-----------------------------------
a12*a15
5*a12*a21*b36 + 5*a1*a21*b36
b59=------------------------------
a12*a15
b58=0
b57=0
b56=0
b55=0
25 2 5 80 5 80 5 2
------*a12 *a15 *b36 - ------*a12*a15 *a1*b36 - ------*a15 *a1 *b36
2187 2187 2187
b54=---------------------------------------------------------------------
7
a12
5 4
----*a15 *b36
81
b53=---------------
4
a12
170 2 4 400 4 400 4 2
-----*a12 *a15 *b36 - -----*a12*a15 *a1*b36 - -----*a15 *a1 *b36
729 729 729
b52=------------------------------------------------------------------
6
a12
10 3
----*a15 *b36
27
b51=---------------
3
a12
10 3
----*a15 *b36
9
b50=---------------
3
a12
65 2 3 100 3 100 3 2
----*a12 *a15 *b36 - -----*a12*a15 *a1*b36 - -----*a15 *a1 *b36
81 81 81
b49=-----------------------------------------------------------------
5
a12
130 2 3 200 3 200 3 2
-----*a12 *a15 *b36 - -----*a12*a15 *a1*b36 - -----*a15 *a1 *b36
243 243 243
b48=------------------------------------------------------------------
5
a12
10 2
----*a15 *b36
9
b47=---------------
2
a12
10 2
----*a15 *b36
3
b46=---------------
2
a12
5 2
---*a15 *b36
3
b45=--------------
2
a12
20 2
----*a15 *b36
9
b44=---------------
2
a12
110 2 2 100 2 100 2 2
-----*a12 *a15 *b36 - -----*a12*a15 *a1*b36 - -----*a15 *a1 *b36
81 81 81
b43=------------------------------------------------------------------
4
a12
55 2 2 50 2 50 2 2
----*a12 *a15 *b36 - ----*a12*a15 *a1*b36 - ----*a15 *a1 *b36
81 81 81
b42=---------------------------------------------------------------
4
a12
5
---*a15*b36
3
b41=-------------
a12
10
----*a15*b36
3
b40=--------------
a12
10
----*a15*b36
3
b39=--------------
a12
5
---*a15*b36
3
b38=-------------
a12
10 2 5 5 2
----*a12 *a15*b36 - ----*a12*a15*a1*b36 - ----*a15*a1 *b36
27 27 27
b37=------------------------------------------------------------
3
a12
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
b28=0
50 40
- ----*a12*a15*a21*b36 - ----*a15*a1*a21*b36
27 27
b27=-----------------------------------------------
3
a12
25 20
- ----*a12*a15*a21*b36 - ----*a15*a1*a21*b36
9 9
b26=-----------------------------------------------
3
a12
25 20
- ----*a12*a21*b36 - ----*a1*a21*b36
3 3
b25=---------------------------------------
2
a12
25 20
- ----*a12*a21*b36 - ----*a1*a21*b36
3 3
b24=---------------------------------------
2
a12
80
- ----*a12*a21*b36 - 20*a1*a21*b36
3
b23=-------------------------------------
2
a12
10
- 5*a12*a21*b36 - ----*a1*a21*b36
3
b22=------------------------------------
2
a12
10
- 5*a12*a21*b36 - ----*a1*a21*b36
3
b21=------------------------------------
2
a12
- 15*a12*a21*b36 - 10*a1*a21*b36
b20=-----------------------------------
a12*a15
- 35*a12*a21*b36 - 30*a1*a21*b36
b19=-----------------------------------
a12*a15
- 30*a12*a21*b36 - 20*a1*a21*b36
b18=-----------------------------------
a12*a15
- 25*a12*a21*b36 - 20*a1*a21*b36
b17=-----------------------------------
a12*a15
65 40
- ----*a12*a21*b36 - ----*a1*a21*b36
3 3
b16=---------------------------------------
a12*a15
10 5
- ----*a12*a21*b36 - ---*a1*a21*b36
3 3
b15=--------------------------------------
a12*a15
- 10*a12*a21*b36 - 5*a1*a21*b36
b14=----------------------------------
2
a15
- 25*a12*a21*b36 - 20*a1*a21*b36
b13=-----------------------------------
2
a15
- 15*a12*a21*b36 - 15*a1*a21*b36
b12=-----------------------------------
2
a15
b11=0
80 2 4 400 4 400 4 2
- -----*a12 *a15 *b36 + -----*a12*a15 *a1*b36 + -----*a15 *a1 *b36
729 729 729
b10=---------------------------------------------------------------------
6
a12
80 2 3 400 3 400 3 2
- ----*a12 *a15 *b36 + -----*a12*a15 *a1*b36 + -----*a15 *a1 *b36
81 81 81
b9=--------------------------------------------------------------------
5
a12
80 2 3 400 3 400 3 2
- -----*a12 *a15 *b36 + -----*a12*a15 *a1*b36 + -----*a15 *a1 *b36
243 243 243
b8=---------------------------------------------------------------------
5
a12
20 2 2 100 2 100 2 2
- ----*a12 *a15 *b36 + -----*a12*a15 *a1*b36 + -----*a15 *a1 *b36
27 27 27
b7=--------------------------------------------------------------------
4
a12
200 2 2 1000 2 1000 2 2
- -----*a12 *a15 *b36 + ------*a12*a15 *a1*b36 + ------*a15 *a1 *b36
81 81 81
b6=-----------------------------------------------------------------------
4
a12
40 2 2 200 2 200 2 2
- ----*a12 *a15 *b36 + -----*a12*a15 *a1*b36 + -----*a15 *a1 *b36
81 81 81
b5=--------------------------------------------------------------------
4
a12
20 2 100 100 2
- ----*a12 *a15*b36 + -----*a12*a15*a1*b36 + -----*a15*a1 *b36
27 27 27
b4=-----------------------------------------------------------------
3
a12
10 2 50 50 2
- ----*a12 *a15*b36 + ----*a12*a15*a1*b36 + ----*a15*a1 *b36
9 9 9
b3=---------------------------------------------------------------
3
a12
10 2 50 50 2
- ----*a12 *a15*b36 + ----*a12*a15*a1*b36 + ----*a15*a1 *b36
27 27 27
b2=---------------------------------------------------------------
3
a12
1 2 5 5 2
- ---*a12 *b36 + ---*a12*a1*b36 + ---*a1 *b36
9 9 9
b1=------------------------------------------------
2
a12
3*a12*a21
a20=-----------
a15
a19=0
1 3 4 3
----*a12*a15 - ----*a15 *a1
27 27
a18=------------------------------
3
a12
1 2
---*a15
3
a17=----------
a12
1 2 2 2
---*a12*a15 - ---*a15 *a1
3 3
a16=----------------------------
2
a12
a14=a15
1 1
---*a12*a15 - ---*a15*a1
3 3
a13=--------------------------
a12
a11=0
a10=0
- 6*a12*a21
a9=--------------
a15
- 3*a12*a21
a8=--------------
a15
2
- 9*a12 *a21
a7=---------------
2
a15
2
- 9*a12 *a21
a6=---------------
2
a15
a5=0
4 2
---*a15 *a1
3
a4=-------------
2
a12
2*a15*a1
a3=----------
a12
2*a15*a1
a2=----------
a12
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:
b36,a21,a15,a12,a1
Relevance for the application:
The solution given above tells us that the system {u_s, v_s}
is a higher order symmetry for the lower order system {u_t,v_t}
where u=u(t,x) is a scalar function, v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 2 3 2 3
u =(u *a12 *a15 *a1 + 2*u *a12*a15 *a1*u + 2*u *a12*a15 *a1
t 3x 2x x
3 4 4 2 4
- 3*u *f(v,v)*a12 *a15*a21 + ---*u *a15 *a1*u - 9*f(v ,v )*a12 *a21
x 3 x x x
4 3 2 2
- 9*f(v,v )*a12 *a21 - 6*f(v,v )*a12 *a15*a21*u)/(a12 *a15 )
2x x
1 3 2 1 2 2 3 2
v =(---*u *a12 *a15 *v - ---*u *a12 *a15 *a1*v + u *v *a12 *a15
t 3 2x 3 2x x x
1 2 3 2 3 4
+ ---*u *a12 *a15 *u*v - ---*u *a12*a15 *a1*u*v + v *a12 *a15
3 x 3 x 3x
3 2 1 2 3 2 4
+ v *a12 *a15 *u + ---*v *a12 *a15 *u + 3*f(v,v )*a12 *a21*v
2x 3 x x
3 1 4 3 4 4 3 3
+ f(v,v)*a12 *a15*a21*u*v + ----*a12*a15 *u *v - ----*a15 *a1*u *v)/(a12
27 27
*a15)
1 6 2 5 5 2
u =( - ---*u *a12 *a15 *b36 + ---*u *a12 *a15 *a1*b36
s 9 5x 9 5x
5 4 2 2 10 5 3
+ ---*u *a12 *a15 *a1 *b36 - ----*u *a12 *a15 *b36*u
9 5x 27 4x
50 4 3 50 3 3 2
+ ----*u *a12 *a15 *a1*b36*u + ----*u *a12 *a15 *a1 *b36*u
27 4x 27 4x
10 5 3 50 4 3
- ----*u *u *a12 *a15 *b36 + ----*u *u *a12 *a15 *a1*b36
9 3x x 9 3x x
50 3 3 2 10 6
+ ----*u *u *a12 *a15 *a1 *b36 - ----*u *f(v,v)*a12 *a15*a21*b36
9 3x x 3 3x
5 5 40 4 4 2
- ---*u *f(v,v)*a12 *a15*a1*a21*b36 - ----*u *a12 *a15 *b36*u
3 3x 81 3x
200 3 4 2 200 2 4 2 2
+ -----*u *a12 *a15 *a1*b36*u + -----*u *a12 *a15 *a1 *b36*u
81 3x 81 3x
20 2 5 3 100 2 4 3
- ----*u *a12 *a15 *b36 + -----*u *a12 *a15 *a1*b36
27 2x 27 2x
100 2 3 3 2 200 4 4
+ -----*u *a12 *a15 *a1 *b36 - -----*u *u *a12 *a15 *b36*u
27 2x 81 2x x
1000 3 4 1000 2 4 2
+ ------*u *u *a12 *a15 *a1*b36*u + ------*u *u *a12 *a15 *a1 *b36*u
81 2x x 81 2x x
65 6 40 5
- ----*u *f(v,v )*a12 *a15*a21*b36 - ----*u *f(v,v )*a12 *a15*a1*a21*b36
3 2x x 3 2x x
5 2 10 4 2
- 5*u *f(v,v)*a12 *a15 *a21*b36*u - ----*u *f(v,v)*a12 *a15 *a1*a21*b36*u
2x 3 2x
80 3 5 3 400 2 5 3
- -----*u *a12 *a15 *b36*u + -----*u *a12 *a15 *a1*b36*u
243 2x 243 2x
400 5 2 3 20 3 4 4
+ -----*u *a12*a15 *a1 *b36*u - ----*u *a12 *a15 *b36
243 2x 27 x
100 3 3 4 100 3 2 4 2
+ -----*u *a12 *a15 *a1*b36 + -----*u *a12 *a15 *a1 *b36
27 x 27 x
2 5 2 10 2 4 2
- 5*u *f(v,v)*a12 *a15 *a21*b36 - ----*u *f(v,v)*a12 *a15 *a1*a21*b36
x 3 x
80 2 3 5 2 400 2 2 5 2
- ----*u *a12 *a15 *b36*u + -----*u *a12 *a15 *a1*b36*u
81 x 81 x
400 2 5 2 2 6
+ -----*u *a12*a15 *a1 *b36*u - 25*u *f(v ,v )*a12 *a15*a21*b36
81 x x x x
5 6
- 20*u *f(v ,v )*a12 *a15*a1*a21*b36 - 30*u *f(v,v )*a12 *a15*a21*b36
x x x x 2x
5 80 5 2
- 20*u *f(v,v )*a12 *a15*a1*a21*b36 - ----*u *f(v,v )*a12 *a15 *a21*b36*u
x 2x 3 x x
4 2
- 20*u *f(v,v )*a12 *a15 *a1*a21*b36*u
x x
25 4 3 2
- ----*u *f(v,v)*a12 *a15 *a21*b36*u
9 x
20 3 3 2 80 2 6 4
- ----*u *f(v,v)*a12 *a15 *a1*a21*b36*u - -----*u *a12 *a15 *b36*u
9 x 729 x
400 6 4 400 6 2 4
+ -----*u *a12*a15 *a1*b36*u + -----*u *a15 *a1 *b36*u
729 x 729 x
7 6
- 15*f(v ,v )*a12 *a21*b36 - 15*f(v ,v )*a12 *a1*a21*b36
2x 2x 2x 2x
7 6
- 25*f(v ,v )*a12 *a21*b36 - 20*f(v ,v )*a12 *a1*a21*b36
x 3x x 3x
6 5
- 35*f(v ,v )*a12 *a15*a21*b36*u - 30*f(v ,v )*a12 *a15*a1*a21*b36*u
x 2x x 2x
25 5 2 2
- ----*f(v ,v )*a12 *a15 *a21*b36*u
3 x x
20 4 2 2 7
- ----*f(v ,v )*a12 *a15 *a1*a21*b36*u - 10*f(v,v )*a12 *a21*b36
3 x x 4x
6 6
- 5*f(v,v )*a12 *a1*a21*b36 - 15*f(v,v )*a12 *a15*a21*b36*u
4x 3x
5 25 5 2 2
- 10*f(v,v )*a12 *a15*a1*a21*b36*u - ----*f(v,v )*a12 *a15 *a21*b36*u
3x 3 2x
20 4 2 2 50 4 3 3
- ----*f(v,v )*a12 *a15 *a1*a21*b36*u - ----*f(v,v )*a12 *a15 *a21*b36*u
3 2x 27 x
40 3 3 3 6 2
- ----*f(v,v )*a12 *a15 *a1*a21*b36*u )/(a12 *a15 )
27 x
10 6 2 5 5 2
v =(----*u *a12 *a15 *b36*v - ----*u *a12 *a15 *a1*b36*v
s 27 4x 27 4x
5 4 2 2 5 6 2
- ----*u *a12 *a15 *a1 *b36*v + ---*u *v *a12 *a15 *b36
27 4x 3 3x x
55 5 3 50 4 3
+ ----*u *a12 *a15 *b36*u*v - ----*u *a12 *a15 *a1*b36*u*v
81 3x 81 3x
50 3 3 2 110 5 3
- ----*u *a12 *a15 *a1 *b36*u*v + -----*u *u *a12 *a15 *b36*v
81 3x 81 2x x
100 4 3 100 3 3 2
- -----*u *u *a12 *a15 *a1*b36*v - -----*u *u *a12 *a15 *a1 *b36*v
81 2x x 81 2x x
10 6 2 20 5 3
+ ----*u *v *a12 *a15 *b36 + ----*u *v *a12 *a15 *b36*u
3 2x 2x 9 2x x
10 6 5 5
+ ----*u *f(v,v)*a12 *a15*a21*b36*v + ---*u *f(v,v)*a12 *a15*a1*a21*b36*v
9 2x 9 2x
130 4 4 2 200 3 4 2
+ -----*u *a12 *a15 *b36*u *v - -----*u *a12 *a15 *a1*b36*u *v
243 2x 243 2x
200 2 4 2 2 5 2 5 3
- -----*u *a12 *a15 *a1 *b36*u *v + ---*u *v *a12 *a15 *b36
243 2x 3 x x
65 2 4 4 100 2 3 4
+ ----*u *a12 *a15 *b36*u*v - -----*u *a12 *a15 *a1*b36*u*v
81 x 81 x
100 2 2 4 2 10 6 2
- -----*u *a12 *a15 *a1 *b36*u*v + ----*u *v *a12 *a15 *b36
81 x 3 x 3x
10 5 3 10 4 4 2
+ ----*u *v *a12 *a15 *b36*u + ----*u *v *a12 *a15 *b36*u
3 x 2x 9 x x
6 10 5
+ 5*u *f(v,v )*a12 *a15*a21*b36*v + ----*u *f(v,v )*a12 *a15*a1*a21*b36*v
x x 3 x x
5 5 2
+ ---*u *f(v,v)*a12 *a15 *a21*b36*u*v
3 x
10 4 2 170 3 5 3
+ ----*u *f(v,v)*a12 *a15 *a1*a21*b36*u*v + -----*u *a12 *a15 *b36*u *v
9 x 729 x
400 2 5 3 400 5 2 3
- -----*u *a12 *a15 *a1*b36*u *v - -----*u *a12*a15 *a1 *b36*u *v
729 x 729 x
7 5 6 2 10 5 3 2
+ v *a12 *a15*b36 + ---*v *a12 *a15 *b36*u + ----*v *a12 *a15 *b36*u
5x 3 4x 9 3x
10 4 4 3 5 3 5 4
+ ----*v *a12 *a15 *b36*u + ----*v *a12 *a15 *b36*u
27 2x 81 x
7 6
+ 5*f(v ,v )*a12 *a21*b36*v + 5*f(v ,v )*a12 *a1*a21*b36*v
x 2x x 2x
10 6
+ ----*f(v ,v )*a12 *a15*a21*b36*u*v
3 x x
10 5 10 7
+ ----*f(v ,v )*a12 *a15*a1*a21*b36*u*v + ----*f(v,v )*a12 *a21*b36*v
3 x x 3 3x
5 6 6
+ ---*f(v,v )*a12 *a1*a21*b36*v + 5*f(v,v )*a12 *a15*a21*b36*u*v
3 3x 2x
10 5
+ ----*f(v,v )*a12 *a15*a1*a21*b36*u*v
3 2x
25 5 2 2
+ ----*f(v,v )*a12 *a15 *a21*b36*u *v
9 x
20 4 2 2
+ ----*f(v,v )*a12 *a15 *a1*a21*b36*u *v
9 x
25 4 3 3
+ ----*f(v,v)*a12 *a15 *a21*b36*u *v
81
20 3 3 3 25 2 6 5
+ ----*f(v,v)*a12 *a15 *a1*a21*b36*u *v + ------*a12 *a15 *b36*u *v
81 2187
80 6 5 80 6 2 5 7
- ------*a12*a15 *a1*b36*u *v - ------*a15 *a1 *b36*u *v)/(a12 *a15)
2187 2187