Solution 2 to problem l12o35
Expressions |
Parameters |
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Back to problem l12o35
Expressions
The solution is given through the following expressions:
b164=0
1 4 1 4
-----*a18*a6 *b1 - -----*a18*a6 *b69
162 162
b163=--------------------------------------
5
a1
b162=0
5 4
----*a6 *b69
81
b161=--------------
4
a1
1 2 3 1 2 3
- ----*a18 *a6 *b1 + ----*a18 *a6 *b69
27 27
b160=-----------------------------------------
5
a1
8 3 8 3
----*a18*a6 *b1 - ----*a18*a6 *b69
27 27
b159=------------------------------------
4
a1
10 3
- ----*a18*a6 *b69
27
b158=---------------------
4
a1
2 3 2 3
----*a18*a6 *b1 - ----*a18*a6 *b69
27 27
b157=------------------------------------
4
a1
b156=0
b155=0
10 3
----*a6 *b69
27
b154=--------------
3
a1
20 3
----*a6 *b69
9
b153=--------------
3
a1
10 3
----*a6 *b69
27
b152=--------------
3
a1
1 3 2 1 3 2
----*a18 *a6 *b1 - ----*a18 *a6 *b69
12 12
b151=--------------------------------------
5
a1
8 2 2 8 2 2
- ---*a18 *a6 *b1 + ---*a18 *a6 *b69
9 9
b150=---------------------------------------
4
a1
5 2 2
---*a18 *a6 *b69
6
b149=------------------
4
a1
7 2 2 7 2 2
- ---*a18 *a6 *b1 + ---*a18 *a6 *b69
9 9
b148=---------------------------------------
4
a1
14 2 14 2
----*a18*a6 *b1 - ----*a18*a6 *b69
9 9
b147=------------------------------------
3
a1
2 2 2 2
---*a18*a6 *b1 - ---*a18*a6 *b69
3 3
b146=----------------------------------
3
a1
2 2 16 2
---*a18*a6 *b1 - ----*a18*a6 *b69
3 9
b145=-----------------------------------
3
a1
20 2
- ----*a18*a6 *b69
3
b144=---------------------
3
a1
5 2
- ---*a18*a6 *b69
3
b143=--------------------
3
a1
16 2 16 2
----*a18*a6 *b1 - ----*a18*a6 *b69
9 9
b142=------------------------------------
3
a1
10 2
- ----*a18*a6 *b69
3
b141=---------------------
3
a1
1 2 4 2
- ---*a18*a6 *b1 - ---*a18*a6 *b69
9 9
b140=-------------------------------------
3
a1
4 2 4 2
---*a18*a6 *b1 - ---*a18*a6 *b69
9 9
b139=----------------------------------
3
a1
b138=0
b137=0
40 2
----*a6 *b69
9
b136=--------------
2
a1
20 2
----*a6 *b69
9
b135=--------------
2
a1
20 2
----*a6 *b69
3
b134=--------------
2
a1
40 2
----*a6 *b69
9
b133=--------------
2
a1
40 2
----*a6 *b69
9
b132=--------------
2
a1
20 2
----*a6 *b69
3
b131=--------------
2
a1
10 2
----*a6 *b69
9
b130=--------------
2
a1
1 4 1 4
- ----*a18 *a6*b1 + ----*a18 *a6*b69
12 12
b129=---------------------------------------
5
a1
2 3 2 3
---*a18 *a6*b1 - ---*a18 *a6*b69
3 3
b128=----------------------------------
4
a1
5 3
- ---*a18 *a6*b69
6
b127=--------------------
4
a1
11 3 11 3
----*a18 *a6*b1 - ----*a18 *a6*b69
6 6
b126=------------------------------------
4
a1
2 2
- a18 *a6*b1 + a18 *a6*b69
b125=-----------------------------
3
a1
2 11 2
- a18 *a6*b1 + ----*a18 *a6*b69
6
b124=----------------------------------
3
a1
2
5*a18 *a6*b69
b123=---------------
3
a1
5 2
---*a18 *a6*b69
2
b122=-----------------
3
a1
22 2 22 2
- ----*a18 *a6*b1 + ----*a18 *a6*b69
3 3
b121=---------------------------------------
3
a1
2
10*a18 *a6*b69
b120=----------------
3
a1
10 2 2
- ----*a18 *a6*b1 + 5*a18 *a6*b69
3
b119=------------------------------------
3
a1
7 2 7 2
- ---*a18 *a6*b1 + ---*a18 *a6*b69
3 3
b118=-------------------------------------
3
a1
2 2
---*a18*a6*b1 - ---*a18*a6*b69
3 3
b117=--------------------------------
2
a1
16
2*a18*a6*b1 - ----*a18*a6*b69
3
b116=-------------------------------
2
a1
20
- ----*a18*a6*b69
3
b115=--------------------
2
a1
20
- ----*a18*a6*b69
3
b114=--------------------
2
a1
- 10*a18*a6*b69
b113=------------------
2
a1
10
- ----*a18*a6*b69
3
b112=--------------------
2
a1
2*a18*a6*b1 - 2*a18*a6*b69
b111=----------------------------
2
a1
26
2*a18*a6*b1 - ----*a18*a6*b69
3
b110=-------------------------------
2
a1
- 20*a18*a6*b69
b109=------------------
2
a1
- 10*a18*a6*b69
b108=------------------
2
a1
14
- 2*a18*a6*b1 - ----*a18*a6*b69
3
b107=----------------------------------
2
a1
20
- ----*a18*a6*b69
3
b106=--------------------
2
a1
14 14
----*a18*a6*b1 - ----*a18*a6*b69
3 3
b105=----------------------------------
2
a1
20
- ----*a18*a6*b69
3
b104=--------------------
2
a1
1
- ---*a18*a6*b1 - 3*a18*a6*b69
3
b103=---------------------------------
2
a1
a18*a6*b1 - a18*a6*b69
b102=------------------------
2
a1
b101=0
10
----*a6*b69
3
b100=-------------
a1
5
---*a6*b69
3
b99=------------
a1
10
----*a6*b69
3
b98=-------------
a1
10*a6*b69
b97=-----------
a1
20
----*a6*b69
3
b96=-------------
a1
20
----*a6*b69
3
b95=-------------
a1
20
----*a6*b69
3
b94=-------------
a1
5
---*a6*b69
3
b93=------------
a1
1 5 1 5
----*a18 *b1 - ----*a18 *b69
32 32
b92=------------------------------
5
a1
5 4
----*a18 *b69
16
b91=---------------
4
a1
5 4 5 4
- ---*a18 *b1 + ---*a18 *b69
4 4
b90=-------------------------------
4
a1
5 3
- ---*a18 *b69
4
b89=-----------------
3
a1
15 3
- ----*a18 *b69
2
b88=------------------
3
a1
35 3 3
----*a18 *b1 - 10*a18 *b69
4
b87=----------------------------
3
a1
5 3 5 3
---*a18 *b1 - ---*a18 *b69
2 2
b86=----------------------------
3
a1
5 2
---*a18 *b69
2
b85=--------------
2
a1
2
15*a18 *b69
b84=-------------
2
a1
2
25*a18 *b69
b83=-------------
2
a1
2
10*a18 *b69
b82=-------------
2
a1
2
10*a18 *b69
b81=-------------
2
a1
25 2 35 2
- ----*a18 *b1 + ----*a18 *b69
2 2
b80=---------------------------------
2
a1
5 2 5 2
- ---*a18 *b1 + ---*a18 *b69
2 2
b79=-------------------------------
2
a1
5
- ---*a18*b69
2
b78=----------------
a1
- 10*a18*b69
b77=---------------
a1
- 10*a18*b69
b76=---------------
a1
- 10*a18*b69
b75=---------------
a1
- 15*a18*b69
b74=---------------
a1
1
---*a18*b1 - 3*a18*b69
2
b73=------------------------
a1
- 5*a18*b69
b72=--------------
a1
- a18*b1 - 4*a18*b69
b71=-----------------------
a1
a18*b1 - a18*b69
b70=------------------
a1
1 5 1 5
-----*a6 *b1 - -----*a6 *b69
243 243
b68=------------------------------
5
a1
2 4 2 4
- ----*a18*a6 *b1 + ----*a18*a6 *b69
81 81
b67=---------------------------------------
5
a1
20 4 20 4
----*a6 *b1 - ----*a6 *b69
81 81
b66=----------------------------
4
a1
5 4
----*a6 *b1
81
b65=-------------
4
a1
1 2 3 1 2 3
----*a18 *a6 *b1 - ----*a18 *a6 *b69
18 18
b64=--------------------------------------
5
a1
22 3 22 3
- ----*a18*a6 *b1 + ----*a18*a6 *b69
27 27
b63=---------------------------------------
4
a1
2 3 8 3
- ---*a18*a6 *b1 + ----*a18*a6 *b69
3 27
b62=--------------------------------------
4
a1
20 3 20 3
----*a6 *b1 - ----*a6 *b69
9 9
b61=----------------------------
3
a1
20 3 20 3
----*a6 *b1 - ----*a6 *b69
27 27
b60=----------------------------
3
a1
20 3 10 3
----*a6 *b1 - ----*a6 *b69
27 27
b59=----------------------------
3
a1
20 3
----*a6 *b1
9
b58=-------------
3
a1
10 3
----*a6 *b1
27
b57=-------------
3
a1
1 3 2 1 3 2
- ----*a18 *a6 *b1 + ----*a18 *a6 *b69
18 18
b56=-----------------------------------------
5
a1
7 2 2 7 2 2
---*a18 *a6 *b1 - ---*a18 *a6 *b69
9 9
b55=------------------------------------
4
a1
31 2 2 8 2 2
----*a18 *a6 *b1 - ---*a18 *a6 *b69
18 9
b54=-------------------------------------
4
a1
16 2 16 2
- ----*a18*a6 *b1 + ----*a18*a6 *b69
9 9
b53=---------------------------------------
3
a1
14 2 14 2
- ----*a18*a6 *b1 + ----*a18*a6 *b69
9 9
b52=---------------------------------------
3
a1
14 2 4 2
- ----*a18*a6 *b1 + ---*a18*a6 *b69
9 9
b51=--------------------------------------
3
a1
104 2 44 2
- -----*a18*a6 *b1 + ----*a18*a6 *b69
9 9
b50=----------------------------------------
3
a1
41 2 2 2
- ----*a18*a6 *b1 + ---*a18*a6 *b69
9 3
b49=--------------------------------------
3
a1
7 2 2 2
- ---*a18*a6 *b1 + ---*a18*a6 *b69
3 3
b48=-------------------------------------
3
a1
40 2 40 2
----*a6 *b1 - ----*a6 *b69
9 9
b47=----------------------------
2
a1
10 2 10 2
----*a6 *b1 - ----*a6 *b69
9 9
b46=----------------------------
2
a1
40 2
----*a6 *b1
9
b45=-------------
2
a1
10 2 10 2
----*a6 *b1 - ----*a6 *b69
3 9
b44=----------------------------
2
a1
20 2
----*a6 *b1
3
b43=-------------
2
a1
40 2
----*a6 *b1
9
b42=-------------
2
a1
40 2
----*a6 *b1
9
b41=-------------
2
a1
20 2
----*a6 *b1
3
b40=-------------
2
a1
10 2
----*a6 *b1
9
b39=-------------
2
a1
1 4 1 4
----*a18 *a6*b1 - ----*a18 *a6*b69
48 48
b38=------------------------------------
5
a1
1 3 1 3
- ---*a18 *a6*b1 + ---*a18 *a6*b69
6 6
b37=-------------------------------------
4
a1
3 3 2 3
- ---*a18 *a6*b1 + ---*a18 *a6*b69
2 3
b36=-------------------------------------
4
a1
2 2 2 2
---*a18 *a6*b1 - ---*a18 *a6*b69
3 3
b35=----------------------------------
3
a1
2 2 1 2
---*a18 *a6*b1 + ---*a18 *a6*b69
3 6
b34=----------------------------------
3
a1
23 2 8 2
----*a18 *a6*b1 - ---*a18 *a6*b69
3 3
b33=-----------------------------------
3
a1
2 10 2
15*a18 *a6*b1 - ----*a18 *a6*b69
3
b32=----------------------------------
3
a1
7 2 2
---*a18 *a6*b1 - a18 *a6*b69
2
b31=------------------------------
3
a1
- a18*a6*b1 + a18*a6*b69
b30=---------------------------
2
a1
1
- 3*a18*a6*b1 - ---*a18*a6*b69
3
b29=---------------------------------
2
a1
34 14
- ----*a18*a6*b1 + ----*a18*a6*b69
3 3
b28=-------------------------------------
2
a1
34
- ----*a18*a6*b1 - 2*a18*a6*b69
3
b27=----------------------------------
2
a1
86
- ----*a18*a6*b1 + 2*a18*a6*b69
3
b26=----------------------------------
2
a1
20
- ----*a18*a6*b1
3
b25=-------------------
2
a1
- 12*a18*a6*b1 + 2*a18*a6*b69
b24=--------------------------------
2
a1
- 22*a18*a6*b1 + 2*a18*a6*b69
b23=--------------------------------
2
a1
2
- 4*a18*a6*b1 + ---*a18*a6*b69
3
b22=---------------------------------
2
a1
2 2
---*a6*b1 - ---*a6*b69
3 3
b21=------------------------
a1
8 2
---*a6*b1 + ---*a6*b69
3 3
b20=------------------------
a1
1
2*a6*b1 - ---*a6*b69
3
b19=----------------------
a1
10
----*a6*b1
3
b18=------------
a1
10*a6*b1
b17=----------
a1
20
----*a6*b1
3
b16=------------
a1
20
----*a6*b1
3
b15=------------
a1
20
----*a6*b1
3
b14=------------
a1
5
---*a6*b1
3
b13=-----------
a1
b12=0
5 4
----*a18 *b1
16
b11=--------------
4
a1
35 3
- ----*a18 *b1
4
b10=-----------------
3
a1
5 3
- ---*a18 *b1
4
b9=----------------
3
a1
2
35*a18 *b1
b8=------------
2
a1
2
30*a18 *b1
b7=------------
2
a1
5 2
---*a18 *b1
2
b6=-------------
2
a1
- 25*a18*b1
b5=--------------
a1
25
- ----*a18*b1
2
b4=----------------
a1
- 20*a18*b1
b3=--------------
a1
5
- ---*a18*b1
2
b2=---------------
a1
a36=0
1 2
----*a18*a6
18
a35=--------------
2
a1
a34=0
a33=0
1 2
- ---*a18 *a6
6
a32=----------------
2
a1
2
---*a18*a6
3
a31=------------
a1
a30=0
1
---*a18*a6
3
a29=------------
a1
a28=0
a27=0
a26=0
a25=0
1 3
---*a18
8
a24=----------
2
a1
a23=0
3 2
- ---*a18
2
a22=-------------
a1
a21=0
a20=0
1
a19= - ---*a18
2
a17=0
1 3
----*a6
27
a16=----------
2
a1
1 2
- ---*a18*a6
9
a15=----------------
2
a1
2 2
---*a6
3
a14=---------
a1
1 2
---*a6
3
a13=---------
a1
1 2
----*a18 *a6
12
a12=--------------
2
a1
1
- ---*a18*a6
3
a11=---------------
a1
5
- ---*a18*a6
3
a10=---------------
a1
2
a9=---*a6
3
2
a8=---*a6
3
a7=2*a6
a5=0
3 2
---*a18
4
a4=----------
a1
9
a3= - ---*a18
2
3
a2= - ---*a18
2
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:
b69,b1,a6,a18,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
3 2 3 2 2 9 2 2
df(u,t)=(u *a1 + u *f(v,v)*a1 *a6 - ---*u *a1 *a18*u - ---*u *a1 *a18*u
3x 2x 2 2x 2 x
2 1 2 2
+ 2*u *f(v,v )*a1 *a6 + ---*u *f(v,v) *a1*a6
x x 3 x
5 2 3 2 4
- ---*u *f(v,v)*a1*a18*a6*u + ---*u *a1*a18 *u
3 x 4 x
2 2 2 2
+ ---*f(v ,v )*a1 *a6*u + ---*f(v,v )*a1 *a6*u
3 x x 3 2x
2 2 1 3
+ ---*f(v,v )*f(v,v)*a1*a6 *u - ---*f(v,v )*a1*a18*a6*u
3 x 3 x
1 3 3 1 2 2 3 1 2 5
+ ----*f(v,v) *a6 *u - ---*f(v,v) *a18*a6 *u + ----*f(v,v)*a18 *a6*u
27 9 12
2
)/a1
5 5 4 5 4 2
df(u,s)=(u *a1 *b1 + ---*u *f(v,v)*a1 *a6*b1 - ---*u *a1 *a18*b1*u
5x 3 4x 2 4x
4 20 4
- 20*u *u *a1 *a18*b1*u + ----*u *f(v,v )*a1 *a6*b1
3x x 3 3x x
10 2 3 2 3 2
+ ----*u *f(v,v) *a1 *a6 *b1 - 4*u *f(v,v)*a1 *a18*a6*b1*u
9 3x 3x
2 3 2 5 3 2 4
+ ---*u *f(v,v)*a1 *a18*a6*b69*u + ---*u *a1 *a18 *b1*u
3 3x 2 3x
25 2 4 2 4
- ----*u *a1 *a18*b1*u - 25*u *u *a1 *a18*b1
2 2x 2x x
3 3
- 22*u *u *f(v,v)*a1 *a18*a6*b1*u + 2*u *u *f(v,v)*a1 *a18*a6*b69*u
2x x 2x x
3 2 3 20 4
+ 30*u *u *a1 *a18 *b1*u + ----*u *f(v ,v )*a1 *a6*b1
2x x 3 2x x x
20 4 20 3 2
+ ----*u *f(v,v )*a1 *a6*b1 + ----*u *f(v,v )*f(v,v)*a1 *a6 *b1
3 2x 2x 3 2x x
3 2 3 2
- 12*u *f(v,v )*a1 *a18*a6*b1*u + 2*u *f(v,v )*a1 *a18*a6*b69*u
2x x 2x x
10 3 2 3 7 2 2 2 2
+ ----*u *f(v,v) *a1 *a6 *b1 - ---*u *f(v,v) *a1 *a18*a6 *b1*u
27 2x 3 2x
2 2 2 2 2
+ ---*u *f(v,v) *a1 *a18*a6 *b69*u
3 2x
7 2 2 4 2 2 4
+ ---*u *f(v,v)*a1 *a18 *a6*b1*u - u *f(v,v)*a1 *a18 *a6*b69*u
2 2x 2x
5 2 3 6 20 3 3
- ---*u *a1 *a18 *b1*u - ----*u *f(v,v)*a1 *a18*a6*b1
4 2x 3 x
3 3 2 2 86 2 3
+ 35*u *a1 *a18 *b1*u - ----*u *f(v,v )*a1 *a18*a6*b1*u
x 3 x x
2 3 41 2 2 2 2
+ 2*u *f(v,v )*a1 *a18*a6*b69*u - ----*u *f(v,v) *a1 *a18*a6 *b1*u
x x 9 x
2 2 2 2 2 2 2 2 3
+ ---*u *f(v,v) *a1 *a18*a6 *b69*u + 15*u *f(v,v)*a1 *a18 *a6*b1*u
3 x x
10 2 2 2 3 35 2 2 3 5
- ----*u *f(v,v)*a1 *a18 *a6*b69*u - ----*u *a1 *a18 *b1*u
3 x 4 x
4 40 3 2
+ 10*u *f(v ,v )*a1 *a6*b1 + ----*u *f(v ,v )*f(v,v)*a1 *a6 *b1
x x 2x 9 x x x
34 3 2 3 2
- ----*u *f(v ,v )*a1 *a18*a6*b1*u - 2*u *f(v ,v )*a1 *a18*a6*b69*u
3 x x x x x x
10 4 40 3 2
+ ----*u *f(v,v )*a1 *a6*b1 + ----*u *f(v,v )*f(v,v)*a1 *a6 *b1
3 x 3x 9 x 2x
34 3 2
- ----*u *f(v,v )*a1 *a18*a6*b1*u
3 x 2x
14 3 2 20 2 3 2
+ ----*u *f(v,v )*a1 *a18*a6*b69*u + ----*u *f(v,v ) *a1 *a6 *b1
3 x 2x 3 x x
20 2 2 3
+ ----*u *f(v,v )*f(v,v) *a1 *a6 *b1
9 x x
104 2 2 2
- -----*u *f(v,v )*f(v,v)*a1 *a18*a6 *b1*u
9 x x
44 2 2 2
+ ----*u *f(v,v )*f(v,v)*a1 *a18*a6 *b69*u
9 x x
23 2 2 4
+ ----*u *f(v,v )*a1 *a18 *a6*b1*u
3 x x
8 2 2 4 5 4 4
- ---*u *f(v,v )*a1 *a18 *a6*b69*u + ----*u *f(v,v) *a1*a6 *b1
3 x x 81 x
2 3 3 2 8 3 3 2
- ---*u *f(v,v) *a1*a18*a6 *b1*u + ----*u *f(v,v) *a1*a18*a6 *b69*u
3 x 27 x
31 2 2 2 4
+ ----*u *f(v,v) *a1*a18 *a6 *b1*u
18 x
8 2 2 2 4 3 3 6
- ---*u *f(v,v) *a1*a18 *a6 *b69*u - ---*u *f(v,v)*a1*a18 *a6*b1*u
9 x 2 x
2 3 6 5 4 8
+ ---*u *f(v,v)*a1*a18 *a6*b69*u + ----*u *a1*a18 *b1*u
3 x 16 x
4 1 4
+ 2*f(v ,v )*a1 *a6*b1*u - ---*f(v ,v )*a1 *a6*b69*u
2x 2x 3 2x 2x
8 4 2 4
+ ---*f(v ,v )*a1 *a6*b1*u + ---*f(v ,v )*a1 *a6*b69*u
3 x 3x 3 x 3x
10 3 2
+ ----*f(v ,v )*f(v,v)*a1 *a6 *b1*u
3 x 2x
10 3 2 3 3
- ----*f(v ,v )*f(v,v)*a1 *a6 *b69*u - 3*f(v ,v )*a1 *a18*a6*b1*u
9 x 2x x 2x
1 3 3 40 3 2
- ---*f(v ,v )*a1 *a18*a6*b69*u + ----*f(v ,v )*f(v,v )*a1 *a6 *b1*u
3 x 2x 9 x x x
20 2 2 3
+ ----*f(v ,v )*f(v,v) *a1 *a6 *b1*u
27 x x
10 2 2 3
- ----*f(v ,v )*f(v,v) *a1 *a6 *b69*u
27 x x
14 2 2 3
- ----*f(v ,v )*f(v,v)*a1 *a18*a6 *b1*u
9 x x
4 2 2 3
+ ---*f(v ,v )*f(v,v)*a1 *a18*a6 *b69*u
9 x x
2 2 2 5 1 2 2 5
+ ---*f(v ,v )*a1 *a18 *a6*b1*u + ---*f(v ,v )*a1 *a18 *a6*b69*u
3 x x 6 x x
2 4 2 4
+ ---*f(v,v )*a1 *a6*b1*u - ---*f(v,v )*a1 *a6*b69*u
3 4x 3 4x
10 3 2
+ ----*f(v,v )*f(v,v)*a1 *a6 *b1*u
9 3x
10 3 2 3 3
- ----*f(v,v )*f(v,v)*a1 *a6 *b69*u - f(v,v )*a1 *a18*a6*b1*u
9 3x 3x
3 3 40 3 2
+ f(v,v )*a1 *a18*a6*b69*u + ----*f(v,v )*f(v,v )*a1 *a6 *b1*u
3x 9 2x x
40 3 2
- ----*f(v,v )*f(v,v )*a1 *a6 *b69*u
9 2x x
20 2 2 3
+ ----*f(v,v )*f(v,v) *a1 *a6 *b1*u
27 2x
20 2 2 3
- ----*f(v,v )*f(v,v) *a1 *a6 *b69*u
27 2x
14 2 2 3
- ----*f(v,v )*f(v,v)*a1 *a18*a6 *b1*u
9 2x
14 2 2 3
+ ----*f(v,v )*f(v,v)*a1 *a18*a6 *b69*u
9 2x
2 2 2 5 2 2 2 5
+ ---*f(v,v )*a1 *a18 *a6*b1*u - ---*f(v,v )*a1 *a18 *a6*b69*u
3 2x 3 2x
20 2 2 3
+ ----*f(v,v ) *f(v,v)*a1 *a6 *b1*u
9 x
20 2 2 3 16 2 2 2 3
- ----*f(v,v ) *f(v,v)*a1 *a6 *b69*u - ----*f(v,v ) *a1 *a18*a6 *b1*u
9 x 9 x
16 2 2 2 3 20 3 4
+ ----*f(v,v ) *a1 *a18*a6 *b69*u + ----*f(v,v )*f(v,v) *a1*a6 *b1*u
9 x 81 x
20 3 4
- ----*f(v,v )*f(v,v) *a1*a6 *b69*u
81 x
22 2 3 3
- ----*f(v,v )*f(v,v) *a1*a18*a6 *b1*u
27 x
22 2 3 3
+ ----*f(v,v )*f(v,v) *a1*a18*a6 *b69*u
27 x
7 2 2 5
+ ---*f(v,v )*f(v,v)*a1*a18 *a6 *b1*u
9 x
7 2 2 5 1 3 7
- ---*f(v,v )*f(v,v)*a1*a18 *a6 *b69*u - ---*f(v,v )*a1*a18 *a6*b1*u
9 x 6 x
1 3 7 1 5 5
+ ---*f(v,v )*a1*a18 *a6*b69*u + -----*f(v,v) *a6 *b1*u
6 x 243
1 5 5 2 4 4 3
- -----*f(v,v) *a6 *b69*u - ----*f(v,v) *a18*a6 *b1*u
243 81
2 4 4 3 1 3 2 3 5
+ ----*f(v,v) *a18*a6 *b69*u + ----*f(v,v) *a18 *a6 *b1*u
81 18
1 3 2 3 5 1 2 3 2 7
- ----*f(v,v) *a18 *a6 *b69*u - ----*f(v,v) *a18 *a6 *b1*u
18 18
1 2 3 2 7 1 4 9
+ ----*f(v,v) *a18 *a6 *b69*u + ----*f(v,v)*a18 *a6*b1*u
18 48
1 4 9 5
- ----*f(v,v)*a18 *a6*b69*u )/a1
48
2 1 2 2 1
df(v,t)=(u *a1 *a18*u*v - ---*u *a1 *a18*v + ---*u *f(v,v)*a1*a18*a6*u*v
2x 2 x 3 x
3 2 3 2 2
- ---*u *a1*a18 *u *v + ---*f(v,v )*a1*a18*a6*u *v
2 x 3 x
1 2 2 2 1 2 4 1 3 6
+ ----*f(v,v) *a18*a6 *u *v - ---*f(v,v)*a18 *a6*u *v + ---*a18 *u *v)
18 6 8
2
/a1
4 4 4
df(v,s)=(u *a1 *a18*b1*u*v - u *a1 *a18*b69*u*v - u *u *a1 *a18*b1*v
4x 4x 3x x
4 4
- 4*u *u *a1 *a18*b69*v - 5*u *v *a1 *a18*b69*u
3x x 3x x
3 3
+ u *f(v,v)*a1 *a18*a6*b1*u*v - u *f(v,v)*a1 *a18*a6*b69*u*v
3x 3x
5 3 2 3 5 3 2 3
- ---*u *a1 *a18 *b1*u *v + ---*u *a1 *a18 *b69*u *v
2 3x 2 3x
1 2 4 2 4
+ ---*u *a1 *a18*b1*v - 3*u *a1 *a18*b69*v
2 2x 2x
4 1 3
- 15*u *u *v *a1 *a18*b69 - ---*u *u *f(v,v)*a1 *a18*a6*b1*v
2x x x 3 2x x
3 25 3 2 2
- 3*u *u *f(v,v)*a1 *a18*a6*b69*v - ----*u *u *a1 *a18 *b1*u *v
2x x 2 2x x
35 3 2 2 4
+ ----*u *u *a1 *a18 *b69*u *v - 10*u *v *a1 *a18*b69*u
2 2x x 2x 2x
20 3 3 2 3
- ----*u *v *f(v,v)*a1 *a18*a6*b69*u + 10*u *v *a1 *a18 *b69*u
3 2x x 2x x
14 3
+ ----*u *f(v,v )*a1 *a18*a6*b1*u*v
3 2x x
14 3
- ----*u *f(v,v )*a1 *a18*a6*b69*u*v
3 2x x
4 2 2 2
+ ---*u *f(v,v) *a1 *a18*a6 *b1*u*v
9 2x
4 2 2 2
- ---*u *f(v,v) *a1 *a18*a6 *b69*u*v
9 2x
7 2 2 3
- ---*u *f(v,v)*a1 *a18 *a6*b1*u *v
3 2x
7 2 2 3 5 2 3 5
+ ---*u *f(v,v)*a1 *a18 *a6*b69*u *v + ---*u *a1 *a18 *b1*u *v
3 2x 2 2x
5 2 3 5 3 3 2
- ---*u *a1 *a18 *b69*u *v + 10*u *a1 *a18 *b69*u*v
2 2x x
2 4 20 2 3
- 10*u *v *a1 *a18*b69 - ----*u *v *f(v,v)*a1 *a18*a6*b69
x 2x 3 x x
2 3 2 2 2 3
+ 25*u *v *a1 *a18 *b69*u - 2*u *f(v,v )*a1 *a18*a6*b1*v
x x x x
14 2 3 1 2 2 2 2
- ----*u *f(v,v )*a1 *a18*a6*b69*v - ---*u *f(v,v) *a1 *a18*a6 *b1*v
3 x x 9 x
4 2 2 2 2
- ---*u *f(v,v) *a1 *a18*a6 *b69*v
9 x
10 2 2 2 2
- ----*u *f(v,v)*a1 *a18 *a6*b1*u *v
3 x
2 2 2 2 35 2 2 3 4
+ 5*u *f(v,v)*a1 *a18 *a6*b69*u *v + ----*u *a1 *a18 *b1*u *v
x 4 x
2 2 3 4 4
- 10*u *a1 *a18 *b69*u *v - 10*u *v *a1 *a18*b69*u
x x 3x
3 3 2 3
- 10*u *v *f(v,v)*a1 *a18*a6*b69*u + 15*u *v *a1 *a18 *b69*u
x 2x x 2x
3
- 20*u *v *f(v,v )*a1 *a18*a6*b69*u
x x x
10 2 2 2
- ----*u *v *f(v,v) *a1 *a18*a6 *b69*u
3 x x
2 2 3 15 2 3 5
+ 10*u *v *f(v,v)*a1 *a18 *a6*b69*u - ----*u *v *a1 *a18 *b69*u
x x 2 x x
3
+ 2*u *f(v ,v )*a1 *a18*a6*b1*u*v
x x x
26 3
- ----*u *f(v ,v )*a1 *a18*a6*b69*u*v
3 x x x
3 3
+ 2*u *f(v,v )*a1 *a18*a6*b1*u*v - 2*u *f(v,v )*a1 *a18*a6*b69*u*v
x 2x x 2x
16 2 2
+ ----*u *f(v,v )*f(v,v)*a1 *a18*a6 *b1*u*v
9 x x
16 2 2
- ----*u *f(v,v )*f(v,v)*a1 *a18*a6 *b69*u*v
9 x x
22 2 2 3
- ----*u *f(v,v )*a1 *a18 *a6*b1*u *v
3 x x
22 2 2 3
+ ----*u *f(v,v )*a1 *a18 *a6*b69*u *v
3 x x
2 3 3
+ ----*u *f(v,v) *a1*a18*a6 *b1*u*v
27 x
2 3 3
- ----*u *f(v,v) *a1*a18*a6 *b69*u*v
27 x
7 2 2 2 3
- ---*u *f(v,v) *a1*a18 *a6 *b1*u *v
9 x
7 2 2 2 3
+ ---*u *f(v,v) *a1*a18 *a6 *b69*u *v
9 x
11 3 5
+ ----*u *f(v,v)*a1*a18 *a6*b1*u *v
6 x
11 3 5 5 4 7
- ----*u *f(v,v)*a1*a18 *a6*b69*u *v - ---*u *a1*a18 *b1*u *v
6 x 4 x
5 4 7 5 5 4
+ ---*u *a1*a18 *b69*u *v + v *a1 *b69 + ---*v *f(v,v)*a1 *a6*b69
4 x 5x 3 4x
5 4 2 20 4
- ---*v *a1 *a18*b69*u + ----*v *f(v,v )*a1 *a6*b69
2 4x 3 3x x
10 2 3 2 10 3 2
+ ----*v *f(v,v) *a1 *a6 *b69 - ----*v *f(v,v)*a1 *a18*a6*b69*u
9 3x 3 3x
5 3 2 4 20 4
+ ---*v *a1 *a18 *b69*u + ----*v *f(v ,v )*a1 *a6*b69
2 3x 3 2x x x
20 4 20 3 2
+ ----*v *f(v,v )*a1 *a6*b69 + ----*v *f(v,v )*f(v,v)*a1 *a6 *b69
3 2x 2x 3 2x x
3 2 10 3 2 3
- 10*v *f(v,v )*a1 *a18*a6*b69*u + ----*v *f(v,v) *a1 *a6 *b69
2x x 27 2x
5 2 2 2 2
- ---*v *f(v,v) *a1 *a18*a6 *b69*u
3 2x
5 2 2 4 5 2 3 6
+ ---*v *f(v,v)*a1 *a18 *a6*b69*u - ---*v *a1 *a18 *b69*u
2 2x 4 2x
4 40 3 2
+ 10*v *f(v ,v )*a1 *a6*b69 + ----*v *f(v ,v )*f(v,v)*a1 *a6 *b69
x x 2x 9 x x x
20 3 2 10 4
- ----*v *f(v ,v )*a1 *a18*a6*b69*u + ----*v *f(v,v )*a1 *a6*b69
3 x x x 3 x 3x
40 3 2
+ ----*v *f(v,v )*f(v,v)*a1 *a6 *b69
9 x 2x
20 3 2 20 2 3 2
- ----*v *f(v,v )*a1 *a18*a6*b69*u + ----*v *f(v,v ) *a1 *a6 *b69
3 x 2x 3 x x
20 2 2 3
+ ----*v *f(v,v )*f(v,v) *a1 *a6 *b69
9 x x
20 2 2 2
- ----*v *f(v,v )*f(v,v)*a1 *a18*a6 *b69*u
3 x x
2 2 4 5 4 4
+ 5*v *f(v,v )*a1 *a18 *a6*b69*u + ----*v *f(v,v) *a1*a6 *b69
x x 81 x
10 3 3 2
- ----*v *f(v,v) *a1*a18*a6 *b69*u
27 x
5 2 2 2 4 5 3 6
+ ---*v *f(v,v) *a1*a18 *a6 *b69*u - ---*v *f(v,v)*a1*a18 *a6*b69*u
6 x 6 x
5 4 8 5 4
+ ----*v *a1*a18 *b69*u + ---*f(v ,v )*a1 *a6*b69*v
16 x 3 2x 2x
10 4 20 3 2
+ ----*f(v ,v )*a1 *a6*b69*v + ----*f(v ,v )*f(v,v)*a1 *a6 *b69*v
3 x 3x 9 x 2x
3 2 16 3 2
+ 2*f(v ,v )*a1 *a18*a6*b1*u *v - ----*f(v ,v )*a1 *a18*a6*b69*u *v
x 2x 3 x 2x
40 3 2
+ ----*f(v ,v )*f(v,v )*a1 *a6 *b69*v
9 x x x
10 2 2 3
+ ----*f(v ,v )*f(v,v) *a1 *a6 *b69*v
27 x x
2 2 2 2
+ ---*f(v ,v )*f(v,v)*a1 *a18*a6 *b1*u *v
3 x x
16 2 2 2
- ----*f(v ,v )*f(v,v)*a1 *a18*a6 *b69*u *v
9 x x
2 2 4 11 2 2 4
- f(v ,v )*a1 *a18 *a6*b1*u *v + ----*f(v ,v )*a1 *a18 *a6*b69*u *v
x x 6 x x
2 3 2 2 3 2
+ ---*f(v,v )*a1 *a18*a6*b1*u *v - ---*f(v,v )*a1 *a18*a6*b69*u *v
3 3x 3 3x
2 2 2 2
+ ---*f(v,v )*f(v,v)*a1 *a18*a6 *b1*u *v
3 2x
2 2 2 2
- ---*f(v,v )*f(v,v)*a1 *a18*a6 *b69*u *v
3 2x
2 2 4 2 2 4
- f(v,v )*a1 *a18 *a6*b1*u *v + f(v,v )*a1 *a18 *a6*b69*u *v
2x 2x
14 2 2 2 2
+ ----*f(v,v ) *a1 *a18*a6 *b1*u *v
9 x
14 2 2 2 2
- ----*f(v,v ) *a1 *a18*a6 *b69*u *v
9 x
8 2 3 2
+ ----*f(v,v )*f(v,v) *a1*a18*a6 *b1*u *v
27 x
8 2 3 2
- ----*f(v,v )*f(v,v) *a1*a18*a6 *b69*u *v
27 x
8 2 2 4
- ---*f(v,v )*f(v,v)*a1*a18 *a6 *b1*u *v
9 x
8 2 2 4
+ ---*f(v,v )*f(v,v)*a1*a18 *a6 *b69*u *v
9 x
2 3 6 2 3 6
+ ---*f(v,v )*a1*a18 *a6*b1*u *v - ---*f(v,v )*a1*a18 *a6*b69*u *v
3 x 3 x
1 4 4 2 1 4 4 2
+ -----*f(v,v) *a18*a6 *b1*u *v - -----*f(v,v) *a18*a6 *b69*u *v
162 162
1 3 2 3 4 1 3 2 3 4
- ----*f(v,v) *a18 *a6 *b1*u *v + ----*f(v,v) *a18 *a6 *b69*u *v
27 27
1 2 3 2 6 1 2 3 2 6
+ ----*f(v,v) *a18 *a6 *b1*u *v - ----*f(v,v) *a18 *a6 *b69*u *v
12 12
1 4 8 1 4 8
- ----*f(v,v)*a18 *a6*b1*u *v + ----*f(v,v)*a18 *a6*b69*u *v
12 12
1 5 10 1 5 10 5
+ ----*a18 *b1*u *v - ----*a18 *b69*u *v)/a1
32 32
df(u,t)=(df(u,x,3)*a1**3 + df(u,x,2)*f(v,v)*a1**2*a6 - 3/2*df(u,x,2)*a1**2*a18*u
**2 - 9/2*df(u,x)**2*a1**2*a18*u + 2*df(u,x)*f(v,df(v,x))*a1**2*a6 + 1/3*df(u,x)
*f(v,v)**2*a1*a6**2 - 5/3*df(u,x)*f(v,v)*a1*a18*a6*u**2 + 3/4*df(u,x)*a1*a18**2*
u**4 + 2/3*f(df(v,x),df(v,x))*a1**2*a6*u + 2/3*f(v,df(v,x,2))*a1**2*a6*u + 2/3*f
(v,df(v,x))*f(v,v)*a1*a6**2*u - 1/3*f(v,df(v,x))*a1*a18*a6*u**3 + 1/27*f(v,v)**3
*a6**3*u - 1/9*f(v,v)**2*a18*a6**2*u**3 + 1/12*f(v,v)*a18**2*a6*u**5)/a1**2$
df(u,s)=(df(u,x,5)*a1**5*b1 + 5/3*df(u,x,4)*f(v,v)*a1**4*a6*b1 - 5/2*df(u,x,4)*
a1**4*a18*b1*u**2 - 20*df(u,x,3)*df(u,x)*a1**4*a18*b1*u + 20/3*df(u,x,3)*f(v,df(
v,x))*a1**4*a6*b1 + 10/9*df(u,x,3)*f(v,v)**2*a1**3*a6**2*b1 - 4*df(u,x,3)*f(v,v)
*a1**3*a18*a6*b1*u**2 + 2/3*df(u,x,3)*f(v,v)*a1**3*a18*a6*b69*u**2 + 5/2*df(u,x,
3)*a1**3*a18**2*b1*u**4 - 25/2*df(u,x,2)**2*a1**4*a18*b1*u - 25*df(u,x,2)*df(u,x
)**2*a1**4*a18*b1 - 22*df(u,x,2)*df(u,x)*f(v,v)*a1**3*a18*a6*b1*u + 2*df(u,x,2)*
df(u,x)*f(v,v)*a1**3*a18*a6*b69*u + 30*df(u,x,2)*df(u,x)*a1**3*a18**2*b1*u**3 +
20/3*df(u,x,2)*f(df(v,x),df(v,x))*a1**4*a6*b1 + 20/3*df(u,x,2)*f(v,df(v,x,2))*a1
**4*a6*b1 + 20/3*df(u,x,2)*f(v,df(v,x))*f(v,v)*a1**3*a6**2*b1 - 12*df(u,x,2)*f(v
,df(v,x))*a1**3*a18*a6*b1*u**2 + 2*df(u,x,2)*f(v,df(v,x))*a1**3*a18*a6*b69*u**2
+ 10/27*df(u,x,2)*f(v,v)**3*a1**2*a6**3*b1 - 7/3*df(u,x,2)*f(v,v)**2*a1**2*a18*
a6**2*b1*u**2 + 2/3*df(u,x,2)*f(v,v)**2*a1**2*a18*a6**2*b69*u**2 + 7/2*df(u,x,2)
*f(v,v)*a1**2*a18**2*a6*b1*u**4 - df(u,x,2)*f(v,v)*a1**2*a18**2*a6*b69*u**4 - 5/
4*df(u,x,2)*a1**2*a18**3*b1*u**6 - 20/3*df(u,x)**3*f(v,v)*a1**3*a18*a6*b1 + 35*
df(u,x)**3*a1**3*a18**2*b1*u**2 - 86/3*df(u,x)**2*f(v,df(v,x))*a1**3*a18*a6*b1*u
+ 2*df(u,x)**2*f(v,df(v,x))*a1**3*a18*a6*b69*u - 41/9*df(u,x)**2*f(v,v)**2*a1**
2*a18*a6**2*b1*u + 2/3*df(u,x)**2*f(v,v)**2*a1**2*a18*a6**2*b69*u + 15*df(u,x)**
2*f(v,v)*a1**2*a18**2*a6*b1*u**3 - 10/3*df(u,x)**2*f(v,v)*a1**2*a18**2*a6*b69*u
**3 - 35/4*df(u,x)**2*a1**2*a18**3*b1*u**5 + 10*df(u,x)*f(df(v,x),df(v,x,2))*a1
**4*a6*b1 + 40/9*df(u,x)*f(df(v,x),df(v,x))*f(v,v)*a1**3*a6**2*b1 - 34/3*df(u,x)
*f(df(v,x),df(v,x))*a1**3*a18*a6*b1*u**2 - 2*df(u,x)*f(df(v,x),df(v,x))*a1**3*
a18*a6*b69*u**2 + 10/3*df(u,x)*f(v,df(v,x,3))*a1**4*a6*b1 + 40/9*df(u,x)*f(v,df(
v,x,2))*f(v,v)*a1**3*a6**2*b1 - 34/3*df(u,x)*f(v,df(v,x,2))*a1**3*a18*a6*b1*u**2
+ 14/3*df(u,x)*f(v,df(v,x,2))*a1**3*a18*a6*b69*u**2 + 20/3*df(u,x)*f(v,df(v,x))
**2*a1**3*a6**2*b1 + 20/9*df(u,x)*f(v,df(v,x))*f(v,v)**2*a1**2*a6**3*b1 - 104/9*
df(u,x)*f(v,df(v,x))*f(v,v)*a1**2*a18*a6**2*b1*u**2 + 44/9*df(u,x)*f(v,df(v,x))*
f(v,v)*a1**2*a18*a6**2*b69*u**2 + 23/3*df(u,x)*f(v,df(v,x))*a1**2*a18**2*a6*b1*u
**4 - 8/3*df(u,x)*f(v,df(v,x))*a1**2*a18**2*a6*b69*u**4 + 5/81*df(u,x)*f(v,v)**4
*a1*a6**4*b1 - 2/3*df(u,x)*f(v,v)**3*a1*a18*a6**3*b1*u**2 + 8/27*df(u,x)*f(v,v)
**3*a1*a18*a6**3*b69*u**2 + 31/18*df(u,x)*f(v,v)**2*a1*a18**2*a6**2*b1*u**4 - 8/
9*df(u,x)*f(v,v)**2*a1*a18**2*a6**2*b69*u**4 - 3/2*df(u,x)*f(v,v)*a1*a18**3*a6*
b1*u**6 + 2/3*df(u,x)*f(v,v)*a1*a18**3*a6*b69*u**6 + 5/16*df(u,x)*a1*a18**4*b1*u
**8 + 2*f(df(v,x,2),df(v,x,2))*a1**4*a6*b1*u - 1/3*f(df(v,x,2),df(v,x,2))*a1**4*
a6*b69*u + 8/3*f(df(v,x),df(v,x,3))*a1**4*a6*b1*u + 2/3*f(df(v,x),df(v,x,3))*a1
**4*a6*b69*u + 10/3*f(df(v,x),df(v,x,2))*f(v,v)*a1**3*a6**2*b1*u - 10/9*f(df(v,x
),df(v,x,2))*f(v,v)*a1**3*a6**2*b69*u - 3*f(df(v,x),df(v,x,2))*a1**3*a18*a6*b1*u
**3 - 1/3*f(df(v,x),df(v,x,2))*a1**3*a18*a6*b69*u**3 + 40/9*f(df(v,x),df(v,x))*f
(v,df(v,x))*a1**3*a6**2*b1*u + 20/27*f(df(v,x),df(v,x))*f(v,v)**2*a1**2*a6**3*b1
*u - 10/27*f(df(v,x),df(v,x))*f(v,v)**2*a1**2*a6**3*b69*u - 14/9*f(df(v,x),df(v,
x))*f(v,v)*a1**2*a18*a6**2*b1*u**3 + 4/9*f(df(v,x),df(v,x))*f(v,v)*a1**2*a18*a6
**2*b69*u**3 + 2/3*f(df(v,x),df(v,x))*a1**2*a18**2*a6*b1*u**5 + 1/6*f(df(v,x),df
(v,x))*a1**2*a18**2*a6*b69*u**5 + 2/3*f(v,df(v,x,4))*a1**4*a6*b1*u - 2/3*f(v,df(
v,x,4))*a1**4*a6*b69*u + 10/9*f(v,df(v,x,3))*f(v,v)*a1**3*a6**2*b1*u - 10/9*f(v,
df(v,x,3))*f(v,v)*a1**3*a6**2*b69*u - f(v,df(v,x,3))*a1**3*a18*a6*b1*u**3 + f(v,
df(v,x,3))*a1**3*a18*a6*b69*u**3 + 40/9*f(v,df(v,x,2))*f(v,df(v,x))*a1**3*a6**2*
b1*u - 40/9*f(v,df(v,x,2))*f(v,df(v,x))*a1**3*a6**2*b69*u + 20/27*f(v,df(v,x,2))
*f(v,v)**2*a1**2*a6**3*b1*u - 20/27*f(v,df(v,x,2))*f(v,v)**2*a1**2*a6**3*b69*u -
14/9*f(v,df(v,x,2))*f(v,v)*a1**2*a18*a6**2*b1*u**3 + 14/9*f(v,df(v,x,2))*f(v,v)
*a1**2*a18*a6**2*b69*u**3 + 2/3*f(v,df(v,x,2))*a1**2*a18**2*a6*b1*u**5 - 2/3*f(v
,df(v,x,2))*a1**2*a18**2*a6*b69*u**5 + 20/9*f(v,df(v,x))**2*f(v,v)*a1**2*a6**3*
b1*u - 20/9*f(v,df(v,x))**2*f(v,v)*a1**2*a6**3*b69*u - 16/9*f(v,df(v,x))**2*a1**
2*a18*a6**2*b1*u**3 + 16/9*f(v,df(v,x))**2*a1**2*a18*a6**2*b69*u**3 + 20/81*f(v,
df(v,x))*f(v,v)**3*a1*a6**4*b1*u - 20/81*f(v,df(v,x))*f(v,v)**3*a1*a6**4*b69*u -
22/27*f(v,df(v,x))*f(v,v)**2*a1*a18*a6**3*b1*u**3 + 22/27*f(v,df(v,x))*f(v,v)**
2*a1*a18*a6**3*b69*u**3 + 7/9*f(v,df(v,x))*f(v,v)*a1*a18**2*a6**2*b1*u**5 - 7/9*
f(v,df(v,x))*f(v,v)*a1*a18**2*a6**2*b69*u**5 - 1/6*f(v,df(v,x))*a1*a18**3*a6*b1*
u**7 + 1/6*f(v,df(v,x))*a1*a18**3*a6*b69*u**7 + 1/243*f(v,v)**5*a6**5*b1*u - 1/
243*f(v,v)**5*a6**5*b69*u - 2/81*f(v,v)**4*a18*a6**4*b1*u**3 + 2/81*f(v,v)**4*
a18*a6**4*b69*u**3 + 1/18*f(v,v)**3*a18**2*a6**3*b1*u**5 - 1/18*f(v,v)**3*a18**2
*a6**3*b69*u**5 - 1/18*f(v,v)**2*a18**3*a6**2*b1*u**7 + 1/18*f(v,v)**2*a18**3*a6
**2*b69*u**7 + 1/48*f(v,v)*a18**4*a6*b1*u**9 - 1/48*f(v,v)*a18**4*a6*b69*u**9)/
a1**5$
df(v,t)=(df(u,x,2)*a1**2*a18*u*v - 1/2*df(u,x)**2*a1**2*a18*v + 1/3*df(u,x)*f(v,
v)*a1*a18*a6*u*v - 3/2*df(u,x)*a1*a18**2*u**3*v + 2/3*f(v,df(v,x))*a1*a18*a6*u**
2*v + 1/18*f(v,v)**2*a18*a6**2*u**2*v - 1/6*f(v,v)*a18**2*a6*u**4*v + 1/8*a18**3
*u**6*v)/a1**2$
df(v,s)=(df(u,x,4)*a1**4*a18*b1*u*v - df(u,x,4)*a1**4*a18*b69*u*v - df(u,x,3)*df
(u,x)*a1**4*a18*b1*v - 4*df(u,x,3)*df(u,x)*a1**4*a18*b69*v - 5*df(u,x,3)*df(v,x)
*a1**4*a18*b69*u + df(u,x,3)*f(v,v)*a1**3*a18*a6*b1*u*v - df(u,x,3)*f(v,v)*a1**3
*a18*a6*b69*u*v - 5/2*df(u,x,3)*a1**3*a18**2*b1*u**3*v + 5/2*df(u,x,3)*a1**3*a18
**2*b69*u**3*v + 1/2*df(u,x,2)**2*a1**4*a18*b1*v - 3*df(u,x,2)**2*a1**4*a18*b69*
v - 15*df(u,x,2)*df(u,x)*df(v,x)*a1**4*a18*b69 - 1/3*df(u,x,2)*df(u,x)*f(v,v)*a1
**3*a18*a6*b1*v - 3*df(u,x,2)*df(u,x)*f(v,v)*a1**3*a18*a6*b69*v - 25/2*df(u,x,2)
*df(u,x)*a1**3*a18**2*b1*u**2*v + 35/2*df(u,x,2)*df(u,x)*a1**3*a18**2*b69*u**2*v
- 10*df(u,x,2)*df(v,x,2)*a1**4*a18*b69*u - 20/3*df(u,x,2)*df(v,x)*f(v,v)*a1**3*
a18*a6*b69*u + 10*df(u,x,2)*df(v,x)*a1**3*a18**2*b69*u**3 + 14/3*df(u,x,2)*f(v,
df(v,x))*a1**3*a18*a6*b1*u*v - 14/3*df(u,x,2)*f(v,df(v,x))*a1**3*a18*a6*b69*u*v
+ 4/9*df(u,x,2)*f(v,v)**2*a1**2*a18*a6**2*b1*u*v - 4/9*df(u,x,2)*f(v,v)**2*a1**2
*a18*a6**2*b69*u*v - 7/3*df(u,x,2)*f(v,v)*a1**2*a18**2*a6*b1*u**3*v + 7/3*df(u,x
,2)*f(v,v)*a1**2*a18**2*a6*b69*u**3*v + 5/2*df(u,x,2)*a1**2*a18**3*b1*u**5*v - 5
/2*df(u,x,2)*a1**2*a18**3*b69*u**5*v + 10*df(u,x)**3*a1**3*a18**2*b69*u*v - 10*
df(u,x)**2*df(v,x,2)*a1**4*a18*b69 - 20/3*df(u,x)**2*df(v,x)*f(v,v)*a1**3*a18*a6
*b69 + 25*df(u,x)**2*df(v,x)*a1**3*a18**2*b69*u**2 - 2*df(u,x)**2*f(v,df(v,x))*
a1**3*a18*a6*b1*v - 14/3*df(u,x)**2*f(v,df(v,x))*a1**3*a18*a6*b69*v - 1/9*df(u,x
)**2*f(v,v)**2*a1**2*a18*a6**2*b1*v - 4/9*df(u,x)**2*f(v,v)**2*a1**2*a18*a6**2*
b69*v - 10/3*df(u,x)**2*f(v,v)*a1**2*a18**2*a6*b1*u**2*v + 5*df(u,x)**2*f(v,v)*
a1**2*a18**2*a6*b69*u**2*v + 35/4*df(u,x)**2*a1**2*a18**3*b1*u**4*v - 10*df(u,x)
**2*a1**2*a18**3*b69*u**4*v - 10*df(u,x)*df(v,x,3)*a1**4*a18*b69*u - 10*df(u,x)*
df(v,x,2)*f(v,v)*a1**3*a18*a6*b69*u + 15*df(u,x)*df(v,x,2)*a1**3*a18**2*b69*u**3
- 20*df(u,x)*df(v,x)*f(v,df(v,x))*a1**3*a18*a6*b69*u - 10/3*df(u,x)*df(v,x)*f(v
,v)**2*a1**2*a18*a6**2*b69*u + 10*df(u,x)*df(v,x)*f(v,v)*a1**2*a18**2*a6*b69*u**
3 - 15/2*df(u,x)*df(v,x)*a1**2*a18**3*b69*u**5 + 2*df(u,x)*f(df(v,x),df(v,x))*a1
**3*a18*a6*b1*u*v - 26/3*df(u,x)*f(df(v,x),df(v,x))*a1**3*a18*a6*b69*u*v + 2*df(
u,x)*f(v,df(v,x,2))*a1**3*a18*a6*b1*u*v - 2*df(u,x)*f(v,df(v,x,2))*a1**3*a18*a6*
b69*u*v + 16/9*df(u,x)*f(v,df(v,x))*f(v,v)*a1**2*a18*a6**2*b1*u*v - 16/9*df(u,x)
*f(v,df(v,x))*f(v,v)*a1**2*a18*a6**2*b69*u*v - 22/3*df(u,x)*f(v,df(v,x))*a1**2*
a18**2*a6*b1*u**3*v + 22/3*df(u,x)*f(v,df(v,x))*a1**2*a18**2*a6*b69*u**3*v + 2/
27*df(u,x)*f(v,v)**3*a1*a18*a6**3*b1*u*v - 2/27*df(u,x)*f(v,v)**3*a1*a18*a6**3*
b69*u*v - 7/9*df(u,x)*f(v,v)**2*a1*a18**2*a6**2*b1*u**3*v + 7/9*df(u,x)*f(v,v)**
2*a1*a18**2*a6**2*b69*u**3*v + 11/6*df(u,x)*f(v,v)*a1*a18**3*a6*b1*u**5*v - 11/6
*df(u,x)*f(v,v)*a1*a18**3*a6*b69*u**5*v - 5/4*df(u,x)*a1*a18**4*b1*u**7*v + 5/4*
df(u,x)*a1*a18**4*b69*u**7*v + df(v,x,5)*a1**5*b69 + 5/3*df(v,x,4)*f(v,v)*a1**4*
a6*b69 - 5/2*df(v,x,4)*a1**4*a18*b69*u**2 + 20/3*df(v,x,3)*f(v,df(v,x))*a1**4*a6
*b69 + 10/9*df(v,x,3)*f(v,v)**2*a1**3*a6**2*b69 - 10/3*df(v,x,3)*f(v,v)*a1**3*
a18*a6*b69*u**2 + 5/2*df(v,x,3)*a1**3*a18**2*b69*u**4 + 20/3*df(v,x,2)*f(df(v,x)
,df(v,x))*a1**4*a6*b69 + 20/3*df(v,x,2)*f(v,df(v,x,2))*a1**4*a6*b69 + 20/3*df(v,
x,2)*f(v,df(v,x))*f(v,v)*a1**3*a6**2*b69 - 10*df(v,x,2)*f(v,df(v,x))*a1**3*a18*
a6*b69*u**2 + 10/27*df(v,x,2)*f(v,v)**3*a1**2*a6**3*b69 - 5/3*df(v,x,2)*f(v,v)**
2*a1**2*a18*a6**2*b69*u**2 + 5/2*df(v,x,2)*f(v,v)*a1**2*a18**2*a6*b69*u**4 - 5/4
*df(v,x,2)*a1**2*a18**3*b69*u**6 + 10*df(v,x)*f(df(v,x),df(v,x,2))*a1**4*a6*b69
+ 40/9*df(v,x)*f(df(v,x),df(v,x))*f(v,v)*a1**3*a6**2*b69 - 20/3*df(v,x)*f(df(v,x
),df(v,x))*a1**3*a18*a6*b69*u**2 + 10/3*df(v,x)*f(v,df(v,x,3))*a1**4*a6*b69 + 40
/9*df(v,x)*f(v,df(v,x,2))*f(v,v)*a1**3*a6**2*b69 - 20/3*df(v,x)*f(v,df(v,x,2))*
a1**3*a18*a6*b69*u**2 + 20/3*df(v,x)*f(v,df(v,x))**2*a1**3*a6**2*b69 + 20/9*df(v
,x)*f(v,df(v,x))*f(v,v)**2*a1**2*a6**3*b69 - 20/3*df(v,x)*f(v,df(v,x))*f(v,v)*a1
**2*a18*a6**2*b69*u**2 + 5*df(v,x)*f(v,df(v,x))*a1**2*a18**2*a6*b69*u**4 + 5/81*
df(v,x)*f(v,v)**4*a1*a6**4*b69 - 10/27*df(v,x)*f(v,v)**3*a1*a18*a6**3*b69*u**2 +
5/6*df(v,x)*f(v,v)**2*a1*a18**2*a6**2*b69*u**4 - 5/6*df(v,x)*f(v,v)*a1*a18**3*
a6*b69*u**6 + 5/16*df(v,x)*a1*a18**4*b69*u**8 + 5/3*f(df(v,x,2),df(v,x,2))*a1**4
*a6*b69*v + 10/3*f(df(v,x),df(v,x,3))*a1**4*a6*b69*v + 20/9*f(df(v,x),df(v,x,2))
*f(v,v)*a1**3*a6**2*b69*v + 2*f(df(v,x),df(v,x,2))*a1**3*a18*a6*b1*u**2*v - 16/3
*f(df(v,x),df(v,x,2))*a1**3*a18*a6*b69*u**2*v + 40/9*f(df(v,x),df(v,x))*f(v,df(v
,x))*a1**3*a6**2*b69*v + 10/27*f(df(v,x),df(v,x))*f(v,v)**2*a1**2*a6**3*b69*v +
2/3*f(df(v,x),df(v,x))*f(v,v)*a1**2*a18*a6**2*b1*u**2*v - 16/9*f(df(v,x),df(v,x)
)*f(v,v)*a1**2*a18*a6**2*b69*u**2*v - f(df(v,x),df(v,x))*a1**2*a18**2*a6*b1*u**4
*v + 11/6*f(df(v,x),df(v,x))*a1**2*a18**2*a6*b69*u**4*v + 2/3*f(v,df(v,x,3))*a1
**3*a18*a6*b1*u**2*v - 2/3*f(v,df(v,x,3))*a1**3*a18*a6*b69*u**2*v + 2/3*f(v,df(v
,x,2))*f(v,v)*a1**2*a18*a6**2*b1*u**2*v - 2/3*f(v,df(v,x,2))*f(v,v)*a1**2*a18*a6
**2*b69*u**2*v - f(v,df(v,x,2))*a1**2*a18**2*a6*b1*u**4*v + f(v,df(v,x,2))*a1**2
*a18**2*a6*b69*u**4*v + 14/9*f(v,df(v,x))**2*a1**2*a18*a6**2*b1*u**2*v - 14/9*f(
v,df(v,x))**2*a1**2*a18*a6**2*b69*u**2*v + 8/27*f(v,df(v,x))*f(v,v)**2*a1*a18*a6
**3*b1*u**2*v - 8/27*f(v,df(v,x))*f(v,v)**2*a1*a18*a6**3*b69*u**2*v - 8/9*f(v,df
(v,x))*f(v,v)*a1*a18**2*a6**2*b1*u**4*v + 8/9*f(v,df(v,x))*f(v,v)*a1*a18**2*a6**
2*b69*u**4*v + 2/3*f(v,df(v,x))*a1*a18**3*a6*b1*u**6*v - 2/3*f(v,df(v,x))*a1*a18
**3*a6*b69*u**6*v + 1/162*f(v,v)**4*a18*a6**4*b1*u**2*v - 1/162*f(v,v)**4*a18*a6
**4*b69*u**2*v - 1/27*f(v,v)**3*a18**2*a6**3*b1*u**4*v + 1/27*f(v,v)**3*a18**2*
a6**3*b69*u**4*v + 1/12*f(v,v)**2*a18**3*a6**2*b1*u**6*v - 1/12*f(v,v)**2*a18**3
*a6**2*b69*u**6*v - 1/12*f(v,v)*a18**4*a6*b1*u**8*v + 1/12*f(v,v)*a18**4*a6*b69*
u**8*v + 1/32*a18**5*b1*u**10*v - 1/32*a18**5*b69*u**10*v)/a1**5$