Solution 1 to problem v1l05o57
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b82=0
b81=0
6
7*a2 *b1
b80=-----------
6
15625*a1
b79=0
b78=0
5
21*a2 *b1
b77=-----------
5
3125*a1
5
42*a2 *b1
b76=-----------
5
625*a1
5
21*a2 *b1
b75=-----------
5
3125*a1
b74=0
b73=0
b72=0
4
336*a2 *b1
b71=------------
4
625*a1
4
84*a2 *b1
b70=-----------
4
625*a1
4
252*a2 *b1
b69=------------
4
125*a1
4
56*a2 *b1
b68=-----------
4
125*a1
4
56*a2 *b1
b67=-----------
4
125*a1
4
84*a2 *b1
b66=-----------
4
125*a1
4
7*a2 *b1
b65=----------
4
125*a1
b64=0
b63=0
b62=0
b61=0
3
588*a2 *b1
b60=------------
3
125*a1
3
252*a2 *b1
b59=------------
3
125*a1
3
252*a2 *b1
b58=------------
3
125*a1
3
672*a2 *b1
b57=------------
3
125*a1
3
98*a2 *b1
b56=-----------
3
125*a1
3
63*a2 *b1
b55=-----------
3
125*a1
3
168*a2 *b1
b54=------------
3
25*a1
3
336*a2 *b1
b53=------------
3
25*a1
3
42*a2 *b1
b52=-----------
3
25*a1
3
336*a2 *b1
b51=------------
3
25*a1
3
126*a2 *b1
b50=------------
3
25*a1
3
252*a2 *b1
b49=------------
3
25*a1
3
84*a2 *b1
b48=-----------
3
25*a1
3
84*a2 *b1
b47=-----------
3
25*a1
3
84*a2 *b1
b46=-----------
3
25*a1
3
7*a2 *b1
b45=----------
3
25*a1
b44=0
b43=0
b42=0
2
84*a2 *b1
b41=-----------
2
25*a1
2
252*a2 *b1
b40=------------
2
25*a1
2
504*a2 *b1
b39=------------
2
25*a1
2
308*a2 *b1
b38=------------
2
25*a1
2
56*a2 *b1
b37=-----------
2
25*a1
2
168*a2 *b1
b36=------------
2
25*a1
2
98*a2 *b1
b35=-----------
2
25*a1
2
56*a2 *b1
b34=-----------
2
5*a1
2
84*a2 *b1
b33=-----------
2
5*a1
2
84*a2 *b1
b32=-----------
2
25*a1
2
112*a2 *b1
b31=------------
2
5*a1
2
56*a2 *b1
b30=-----------
2
5*a1
2
252*a2 *b1
b29=------------
2
5*a1
2
336*a2 *b1
b28=------------
2
25*a1
2
252*a2 *b1
b27=------------
2
25*a1
2
168*a2 *b1
b26=------------
2
5*a1
2
42*a2 *b1
b25=-----------
2
5*a1
2
168*a2 *b1
b24=------------
2
5*a1
2
126*a2 *b1
b23=------------
2
5*a1
2
84*a2 *b1
b22=-----------
2
5*a1
2
56*a2 *b1
b21=-----------
2
5*a1
2
56*a2 *b1
b20=-----------
2
5*a1
2
42*a2 *b1
b19=-----------
2
5*a1
2
21*a2 *b1
b18=-----------
2
25*a1
b17=0
14*a2*b1
b16=----------
5*a1
28*a2*b1
b15=----------
5*a1
21*a2*b1
b14=----------
5*a1
14*a2*b1
b13=----------
5*a1
14*a2*b1
b12=----------
a1
28*a2*b1
b11=----------
a1
42*a2*b1
b10=----------
5*a1
168*a2*b1
b9=-----------
5*a1
126*a2*b1
b8=-----------
5*a1
14*a2*b1
b7=----------
a1
42*a2*b1
b6=----------
a1
14*a2*b1
b5=----------
a1
14*a2*b1
b4=----------
a1
42*a2*b1
b3=----------
5*a1
7
---*a2*b1
5
b2=-----------
a1
a27=0
a26=0
4
a2
a25=---------
3
125*a1
a24=0
a23=0
3
2*a2
a22=--------
2
25*a1
3
12*a2
a21=--------
2
25*a1
3
2*a2
a20=--------
2
25*a1
a19=0
a18=0
2
8*a2
a17=-------
5*a1
2
4*a2
a16=-------
5*a1
2
12*a2
a15=--------
5*a1
2
8*a2
a14=-------
5*a1
2
8*a2
a13=-------
5*a1
2
12*a2
a12=--------
5*a1
2
2*a2
a11=-------
5*a1
a10=0
a9=2*a2
a8=a2
a7=2*a2
a6=6*a2
a5=4*a2
a4=4*a2
a3=4*a2
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:
b1,a2,a1
Relevance for the application:
The solution given above tells us that the system v_s
is a higher order symmetry for v_t
where v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
4 3 3
df(v,t)=(v *a1 + v *f(v,v)*a1 *a2 + 4*v *f(v,v )*a1 *a2
5x 4x 3x x
2 2 2 2 3
+ ---*v *f(v,v) *a1 *a2 + 4*v *f(v ,v )*a1 *a2
5 3x 2x x x
3 12 2 2
+ 4*v *f(v,v )*a1 *a2 + ----*v *f(v,v )*f(v,v)*a1 *a2
2x 2x 5 2x x
2 3 3 3
+ ----*v *f(v,v) *a1*a2 + 6*v *f(v ,v )*a1 *a2
25 2x x x 2x
8 2 2 3
+ ---*v *f(v ,v )*f(v,v)*a1 *a2 + 2*v *f(v,v )*a1 *a2
5 x x x x 3x
8 2 2 12 2 2 2
+ ---*v *f(v,v )*f(v,v)*a1 *a2 + ----*v *f(v,v ) *a1 *a2
5 x 2x 5 x x
12 2 3 1 4 4
+ ----*v *f(v,v )*f(v,v) *a1*a2 + -----*v *f(v,v) *a2
25 x x 125 x
3 3
+ f(v ,v )*a1 *a2*v + 2*f(v ,v )*a1 *a2*v
2x 2x x 3x
4 2 2 8 2 2
+ ---*f(v ,v )*f(v,v)*a1 *a2 *v + ---*f(v ,v )*f(v,v )*a1 *a2 *v
5 x 2x 5 x x x
2 2 3 3
+ ----*f(v ,v )*f(v,v) *a1*a2 *v)/a1
25 x x
6 7 5 42 5
df(v,s)=(v *a1 *b1 + ---*v *f(v,v)*a1 *a2*b1 + ----*v *f(v,v )*a1 *a2*b1
7x 5 6x 5 5x x
21 2 4 2 5
+ ----*v *f(v,v) *a1 *a2 *b1 + 14*v *f(v ,v )*a1 *a2*b1
25 5x 4x x x
5 42 4 2
+ 14*v *f(v,v )*a1 *a2*b1 + ----*v *f(v,v )*f(v,v)*a1 *a2 *b1
4x 2x 5 4x x
7 3 3 3 5
+ ----*v *f(v,v) *a1 *a2 *b1 + 42*v *f(v ,v )*a1 *a2*b1
25 4x 3x x 2x
56 4 2 5
+ ----*v *f(v ,v )*f(v,v)*a1 *a2 *b1 + 14*v *f(v,v )*a1 *a2*b1
5 3x x x 3x 3x
56 4 2 84 2 4 2
+ ----*v *f(v,v )*f(v,v)*a1 *a2 *b1 + ----*v *f(v,v ) *a1 *a2 *b1
5 3x 2x 5 3x x
84 2 3 3 7 4 2 4
+ ----*v *f(v,v )*f(v,v) *a1 *a2 *b1 + -----*v *f(v,v) *a1 *a2 *b1
25 3x x 125 3x
126 5 168 5
+ -----*v *f(v ,v )*a1 *a2*b1 + -----*v *f(v ,v )*a1 *a2*b1
5 2x 2x 2x 5 2x x 3x
126 4 2
+ -----*v *f(v ,v )*f(v,v)*a1 *a2 *b1
5 2x x 2x
168 4 2
+ -----*v *f(v ,v )*f(v,v )*a1 *a2 *b1
5 2x x x x
84 2 3 3 42 5
+ ----*v *f(v ,v )*f(v,v) *a1 *a2 *b1 + ----*v *f(v,v )*a1 *a2*b1
25 2x x x 5 2x 4x
42 4 2
+ ----*v *f(v,v )*f(v,v)*a1 *a2 *b1
5 2x 3x
168 4 2
+ -----*v *f(v,v )*f(v,v )*a1 *a2 *b1
5 2x 2x x
84 2 3 3
+ ----*v *f(v,v )*f(v,v) *a1 *a2 *b1
25 2x 2x
252 2 3 3
+ -----*v *f(v,v ) *f(v,v)*a1 *a2 *b1
25 2x x
84 3 2 4 21 5 5
+ -----*v *f(v,v )*f(v,v) *a1 *a2 *b1 + ------*v *f(v,v) *a1*a2 *b1
125 2x x 3125 2x
5 252 4 2
+ 28*v *f(v ,v )*a1 *a2*b1 + -----*v *f(v ,v )*f(v,v)*a1 *a2 *b1
x 2x 3x 25 x 2x 2x
5 336 4 2
+ 14*v *f(v ,v )*a1 *a2*b1 + -----*v *f(v ,v )*f(v,v)*a1 *a2 *b1
x x 4x 25 x x 3x
252 4 2
+ -----*v *f(v ,v )*f(v,v )*a1 *a2 *b1
5 x x 2x x
126 2 3 3
+ -----*v *f(v ,v )*f(v,v) *a1 *a2 *b1
25 x x 2x
56 2 4 2
+ ----*v *f(v ,v ) *a1 *a2 *b1
5 x x x
112 4 2
+ -----*v *f(v ,v )*f(v,v )*a1 *a2 *b1
5 x x x 2x
336 3 3
+ -----*v *f(v ,v )*f(v,v )*f(v,v)*a1 *a2 *b1
25 x x x x
56 3 2 4 14 5
+ -----*v *f(v ,v )*f(v,v) *a1 *a2 *b1 + ----*v *f(v,v )*a1 *a2*b1
125 x x x 5 x 5x
84 4 2
+ ----*v *f(v,v )*f(v,v)*a1 *a2 *b1
25 x 4x
84 4 2
+ ----*v *f(v,v )*f(v,v )*a1 *a2 *b1
5 x 3x x
42 2 3 3 56 2 4 2
+ ----*v *f(v,v )*f(v,v) *a1 *a2 *b1 + ----*v *f(v,v ) *a1 *a2 *b1
25 x 3x 5 x 2x
336 3 3
+ -----*v *f(v,v )*f(v,v )*f(v,v)*a1 *a2 *b1
25 x 2x x
56 3 2 4 168 3 3 3
+ -----*v *f(v,v )*f(v,v) *a1 *a2 *b1 + -----*v *f(v,v ) *a1 *a2 *b1
125 x 2x 25 x x
252 2 2 2 4
+ -----*v *f(v,v ) *f(v,v) *a1 *a2 *b1
125 x x
42 4 5 7 6 6
+ -----*v *f(v,v )*f(v,v) *a1*a2 *b1 + -------*v *f(v,v) *a2 *b1
625 x x 15625 x
21 5 28 5
+ ----*f(v ,v )*a1 *a2*b1*v + ----*f(v ,v )*a1 *a2*b1*v
5 3x 3x 5 2x 4x
98 4 2
+ ----*f(v ,v )*f(v,v)*a1 *a2 *b1*v
25 2x 3x
168 4 2
+ -----*f(v ,v )*f(v,v )*a1 *a2 *b1*v
25 2x 2x x
63 2 3 3 14 5
+ -----*f(v ,v )*f(v,v) *a1 *a2 *b1*v + ----*f(v ,v )*a1 *a2*b1*v
125 2x 2x 5 x 5x
56 4 2
+ ----*f(v ,v )*f(v,v)*a1 *a2 *b1*v
25 x 4x
308 4 2
+ -----*f(v ,v )*f(v,v )*a1 *a2 *b1*v
25 x 3x x
98 2 3 3
+ -----*f(v ,v )*f(v,v) *a1 *a2 *b1*v
125 x 3x
504 4 2
+ -----*f(v ,v )*f(v ,v )*a1 *a2 *b1*v
25 x 2x x x
252 4 2
+ -----*f(v ,v )*f(v,v )*a1 *a2 *b1*v
25 x 2x 2x
672 3 3
+ -----*f(v ,v )*f(v,v )*f(v,v)*a1 *a2 *b1*v
125 x 2x x
84 3 2 4
+ -----*f(v ,v )*f(v,v) *a1 *a2 *b1*v
625 x 2x
252 2 3 3
+ -----*f(v ,v ) *f(v,v)*a1 *a2 *b1*v
125 x x
84 4 2
+ ----*f(v ,v )*f(v,v )*a1 *a2 *b1*v
25 x x 3x
252 3 3
+ -----*f(v ,v )*f(v,v )*f(v,v)*a1 *a2 *b1*v
125 x x 2x
588 2 3 3
+ -----*f(v ,v )*f(v,v ) *a1 *a2 *b1*v
125 x x x
336 2 2 4
+ -----*f(v ,v )*f(v,v )*f(v,v) *a1 *a2 *b1*v
625 x x x
21 4 5 6
+ ------*f(v ,v )*f(v,v) *a1*a2 *b1*v)/a1
3125 x x