Solution 1 to problem l1o35
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
b75=0
b74=0
b73=0
160 4
- -----*a3 *a7*b37
243
b72=---------------------
5
a13
80 3
- ----*a3 *a7*b37
27
b71=--------------------
4
a13
b70=0
140 3
- -----*a3 *a7*b37
81
b69=---------------------
4
a13
70 2
- ----*a3 *a7*b37
27
b68=--------------------
3
a13
50 2
- ----*a3 *a7*b37
27
b67=--------------------
3
a13
b66=0
b65=0
70 2
- ----*a3 *a7*b37
27
b64=--------------------
3
a13
b63=0
5 2
- ---*a3 *a7*b37
9
b62=-------------------
3
a13
5
- ---*a3*a7*b37
6
b61=------------------
2
a13
25
- ----*a3*a7*b37
18
b60=-------------------
2
a13
b59=0
b58=0
b57=0
b56=0
b55=0
80 4
----*a3 *b37
81
b54=--------------
4
a13
80 4
----*a3 *b37
81
b53=--------------
4
a13
80 3
----*a3 *b37
27
b52=--------------
3
a13
80 3
----*a3 *b37
9
b51=--------------
3
a13
10 3
----*a3 *b37
3
b50=--------------
3
a13
20 3
----*a3 *b37
9
b49=--------------
3
a13
40 2
----*a3 *b37
9
b48=--------------
2
a13
40 2
----*a3 *b37
3
b47=--------------
2
a13
20 2
----*a3 *b37
3
b46=--------------
2
a13
80 2
----*a3 *b37
9
b45=--------------
2
a13
35 2
----*a3 *b37
9
b44=--------------
2
a13
35 2
----*a3 *b37
18
b43=--------------
2
a13
10
----*a3*b37
3
b42=-------------
a13
20
----*a3*b37
3
b41=-------------
a13
20
----*a3*b37
3
b40=-------------
a13
10
----*a3*b37
3
b39=-------------
a13
5
---*a3*b37
8
b38=------------
a13
b36=0
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
160 3
-----*a3 *a7*b37
81
b28=------------------
4
a13
80 3
----*a3 *a7*b37
27
b27=-----------------
4
a13
40 2
----*a3 *a7*b37
9
b26=-----------------
3
a13
40 2
----*a3 *a7*b37
9
b25=-----------------
3
a13
380 2
-----*a3 *a7*b37
27
b24=------------------
3
a13
70 2
----*a3 *a7*b37
27
b23=-----------------
3
a13
70 2
----*a3 *a7*b37
27
b22=-----------------
3
a13
35
----*a3*a7*b37
9
b21=----------------
2
a13
85
----*a3*a7*b37
9
b20=----------------
2
a13
70
----*a3*a7*b37
9
b19=----------------
2
a13
20
----*a3*a7*b37
3
b18=----------------
2
a13
50
----*a3*a7*b37
9
b17=----------------
2
a13
5
---*a3*a7*b37
6
b16=---------------
2
a13
5
---*a7*b37
4
b15=------------
a13
10
----*a7*b37
3
b14=-------------
a13
25
----*a7*b37
12
b13=-------------
a13
b12=0
80 4
----*a3 *b37
81
b11=--------------
4
a13
40 3
----*a3 *b37
9
b10=--------------
3
a13
40 3
----*a3 *b37
27
b9=--------------
3
a13
5 2
---*a3 *b37
3
b8=-------------
2
a13
50 2
----*a3 *b37
9
b7=--------------
2
a13
10 2
----*a3 *b37
9
b6=--------------
2
a13
5
---*a3*b37
6
b5=------------
a13
5
---*a3*b37
4
b4=------------
a13
5
----*a3*b37
12
b3=-------------
a13
1
b2=----*b37
16
4 2
- ---*a3 *a7
9
a22=---------------
2
a13
2
- ---*a3*a7
3
a21=--------------
a13
a20=0
a19=0
4 2
---*a3
3
a18=---------
a13
2 2
---*a3
3
a17=---------
a13
a16=2*a3
a15=2*a3
1
a14=---*a3
2
a12=0
a11=0
4
---*a3*a7
3
a10=-----------
a13
2
---*a3*a7
3
a9=-----------
a13
a8=a7
a6=0
4 2
---*a3
3
a5=---------
a13
a4=a3
1
a2=---*a13
4
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:
b37,a7,a3,a13
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1 2 2 2
df(u,t)=(---*u *a13 + u *a13*a3*u + u *a13*a3 + ---*u *f(v,v)*a3*a7
4 3x 2x x 3 x
4 2 2
+ ---*u *a3 *u + f(v ,v )*a13*a7 + f(v,v )*a13*a7
3 x x x 2x
4
+ ---*f(v,v )*a3*a7*u)/a13
3 x
1 2 2 2 2 3
df(v,t)=(---*u *a13 *a3*v + 2*u *v *a13 *a3 + ---*u *a13*a3 *u*v + v *a13
2 2x x x 3 x 3x
2 4 2 2 2
+ 2*v *a13 *a3*u + ---*v *a13*a3 *u - ---*f(v,v )*a13*a3*a7*v
2x 3 x 3 x
4 2 2
- ---*f(v,v)*a3 *a7*u*v)/a13
9
1 4 5 3 5 3
df(u,s)=(----*u *a13 *b37 + ----*u *a13 *a3*b37*u + ---*u *u *a13 *a3*b37
16 5x 12 4x 4 3x x
5 2 10 2 2 2
+ ---*u *f(v,v)*a13 *a3*a7*b37 + ----*u *a13 *a3 *b37*u
6 3x 9 3x
5 2 3 50 2 2
+ ---*u *a13 *a3*b37 + ----*u *u *a13 *a3 *b37*u
6 2x 9 2x x
50 2 70 2
+ ----*u *f(v,v )*a13 *a3*a7*b37 + ----*u *f(v,v)*a13*a3 *a7*b37*u
9 2x x 27 2x
40 3 3 5 3 2 2
+ ----*u *a13*a3 *b37*u + ---*u *a13 *a3 *b37
27 2x 3 x
70 2 2 40 2 3 2
+ ----*u *f(v,v)*a13*a3 *a7*b37 + ----*u *a13*a3 *b37*u
27 x 9 x
20 2 70 2
+ ----*u *f(v ,v )*a13 *a3*a7*b37 + ----*u *f(v,v )*a13 *a3*a7*b37
3 x x x 9 x 2x
380 2 80 3 2
+ -----*u *f(v,v )*a13*a3 *a7*b37*u + ----*u *f(v,v)*a3 *a7*b37*u
27 x x 27 x
80 4 4 25 3
+ ----*u *a3 *b37*u + ----*f(v ,v )*a13 *a7*b37
81 x 12 2x 2x
10 3 85 2
+ ----*f(v ,v )*a13 *a7*b37 + ----*f(v ,v )*a13 *a3*a7*b37*u
3 x 3x 9 x 2x
40 2 2 5 3
+ ----*f(v ,v )*a13*a3 *a7*b37*u + ---*f(v,v )*a13 *a7*b37
9 x x 4 4x
35 2 40 2 2
+ ----*f(v,v )*a13 *a3*a7*b37*u + ----*f(v,v )*a13*a3 *a7*b37*u
9 3x 9 2x
160 3 3 4
+ -----*f(v,v )*a3 *a7*b37*u )/a13
81 x
5 4 10 4
df(v,s)=(---*u *a13 *a3*b37*v + ----*u *v *a13 *a3*b37
8 4x 3 3x x
35 3 2 35 3 2
+ ----*u *a13 *a3 *b37*u*v + ----*u *u *a13 *a3 *b37*v
18 3x 9 2x x
20 4 80 3 2
+ ----*u *v *a13 *a3*b37 + ----*u *v *a13 *a3 *b37*u
3 2x 2x 9 2x x
5 2 2 20 2 3 2
- ---*u *f(v,v)*a13 *a3 *a7*b37*v + ----*u *a13 *a3 *b37*u *v
9 2x 9 2x
20 2 3 2 10 2 2 3
+ ----*u *v *a13 *a3 *b37 + ----*u *a13 *a3 *b37*u*v
3 x x 3 x
20 4 40 3 2
+ ----*u *v *a13 *a3*b37 + ----*u *v *a13 *a3 *b37*u
3 x 3x 3 x 2x
80 2 3 2 70 2 2
+ ----*u *v *a13 *a3 *b37*u - ----*u *f(v,v )*a13 *a3 *a7*b37*v
9 x x 27 x x
140 3 80 4 3
- -----*u *f(v,v)*a13*a3 *a7*b37*u*v + ----*u *a13*a3 *b37*u *v
81 x 81 x
5 10 4 40 3 2 2
+ v *a13 *b37 + ----*v *a13 *a3*b37*u + ----*v *a13 *a3 *b37*u
5x 3 4x 9 3x
80 2 3 3 80 4 4
+ ----*v *a13 *a3 *b37*u + ----*v *a13*a3 *b37*u
27 2x 81 x
25 3 50 2 2
- ----*f(v ,v )*a13 *a3*a7*b37*v - ----*f(v ,v )*a13 *a3 *a7*b37*u*v
18 x 2x 27 x x
5 3 70 2 2
- ---*f(v,v )*a13 *a3*a7*b37*v - ----*f(v,v )*a13 *a3 *a7*b37*u*v
6 3x 27 2x
80 3 2 160 4 3
- ----*f(v,v )*a13*a3 *a7*b37*u *v - -----*f(v,v)*a3 *a7*b37*u *v)/
27 x 243
5
a13
df(u,t)=(1/4*df(u,x,3)*a13**2 + df(u,x,2)*a13*a3*u + df(u,x)**2*a13*a3 + 2/3*df(
u,x)*f(v,v)*a3*a7 + 4/3*df(u,x)*a3**2*u**2 + f(df(v,x),df(v,x))*a13*a7 + f(v,df(
v,x,2))*a13*a7 + 4/3*f(v,df(v,x))*a3*a7*u)/a13$
df(v,t)=(1/2*df(u,x,2)*a13**2*a3*v + 2*df(u,x)*df(v,x)*a13**2*a3 + 2/3*df(u,x)*
a13*a3**2*u*v + df(v,x,3)*a13**3 + 2*df(v,x,2)*a13**2*a3*u + 4/3*df(v,x)*a13*a3
**2*u**2 - 2/3*f(v,df(v,x))*a13*a3*a7*v - 4/9*f(v,v)*a3**2*a7*u*v)/a13**2$
df(u,s)=(1/16*df(u,x,5)*a13**4*b37 + 5/12*df(u,x,4)*a13**3*a3*b37*u + 5/4*df(u,x
,3)*df(u,x)*a13**3*a3*b37 + 5/6*df(u,x,3)*f(v,v)*a13**2*a3*a7*b37 + 10/9*df(u,x,
3)*a13**2*a3**2*b37*u**2 + 5/6*df(u,x,2)**2*a13**3*a3*b37 + 50/9*df(u,x,2)*df(u,
x)*a13**2*a3**2*b37*u + 50/9*df(u,x,2)*f(v,df(v,x))*a13**2*a3*a7*b37 + 70/27*df(
u,x,2)*f(v,v)*a13*a3**2*a7*b37*u + 40/27*df(u,x,2)*a13*a3**3*b37*u**3 + 5/3*df(u
,x)**3*a13**2*a3**2*b37 + 70/27*df(u,x)**2*f(v,v)*a13*a3**2*a7*b37 + 40/9*df(u,x
)**2*a13*a3**3*b37*u**2 + 20/3*df(u,x)*f(df(v,x),df(v,x))*a13**2*a3*a7*b37 + 70/
9*df(u,x)*f(v,df(v,x,2))*a13**2*a3*a7*b37 + 380/27*df(u,x)*f(v,df(v,x))*a13*a3**
2*a7*b37*u + 80/27*df(u,x)*f(v,v)*a3**3*a7*b37*u**2 + 80/81*df(u,x)*a3**4*b37*u
**4 + 25/12*f(df(v,x,2),df(v,x,2))*a13**3*a7*b37 + 10/3*f(df(v,x),df(v,x,3))*a13
**3*a7*b37 + 85/9*f(df(v,x),df(v,x,2))*a13**2*a3*a7*b37*u + 40/9*f(df(v,x),df(v,
x))*a13*a3**2*a7*b37*u**2 + 5/4*f(v,df(v,x,4))*a13**3*a7*b37 + 35/9*f(v,df(v,x,3
))*a13**2*a3*a7*b37*u + 40/9*f(v,df(v,x,2))*a13*a3**2*a7*b37*u**2 + 160/81*f(v,
df(v,x))*a3**3*a7*b37*u**3)/a13**4$
df(v,s)=(5/8*df(u,x,4)*a13**4*a3*b37*v + 10/3*df(u,x,3)*df(v,x)*a13**4*a3*b37 +
35/18*df(u,x,3)*a13**3*a3**2*b37*u*v + 35/9*df(u,x,2)*df(u,x)*a13**3*a3**2*b37*v
+ 20/3*df(u,x,2)*df(v,x,2)*a13**4*a3*b37 + 80/9*df(u,x,2)*df(v,x)*a13**3*a3**2*
b37*u - 5/9*df(u,x,2)*f(v,v)*a13**2*a3**2*a7*b37*v + 20/9*df(u,x,2)*a13**2*a3**3
*b37*u**2*v + 20/3*df(u,x)**2*df(v,x)*a13**3*a3**2*b37 + 10/3*df(u,x)**2*a13**2*
a3**3*b37*u*v + 20/3*df(u,x)*df(v,x,3)*a13**4*a3*b37 + 40/3*df(u,x)*df(v,x,2)*
a13**3*a3**2*b37*u + 80/9*df(u,x)*df(v,x)*a13**2*a3**3*b37*u**2 - 70/27*df(u,x)*
f(v,df(v,x))*a13**2*a3**2*a7*b37*v - 140/81*df(u,x)*f(v,v)*a13*a3**3*a7*b37*u*v
+ 80/81*df(u,x)*a13*a3**4*b37*u**3*v + df(v,x,5)*a13**5*b37 + 10/3*df(v,x,4)*a13
**4*a3*b37*u + 40/9*df(v,x,3)*a13**3*a3**2*b37*u**2 + 80/27*df(v,x,2)*a13**2*a3
**3*b37*u**3 + 80/81*df(v,x)*a13*a3**4*b37*u**4 - 25/18*f(df(v,x),df(v,x,2))*a13
**3*a3*a7*b37*v - 50/27*f(df(v,x),df(v,x))*a13**2*a3**2*a7*b37*u*v - 5/6*f(v,df(
v,x,3))*a13**3*a3*a7*b37*v - 70/27*f(v,df(v,x,2))*a13**2*a3**2*a7*b37*u*v - 80/
27*f(v,df(v,x))*a13*a3**3*a7*b37*u**2*v - 160/243*f(v,v)*a3**4*a7*b37*u**3*v)/
a13**5$