Solution 3 to problem l1o35
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Parameters |
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Expressions
The solution is given through the following expressions:
b75=0
b74=0
b73=0
b72=(1/9*a22*a3**2*b2)/a2**3
b71=(2*a22*a3*b2)/a2**2
b70=0
b69=(a22*a3*b2)/a2**2
b68=(6*a22*b2)/a2
b67=(6*a22*b2)/a2
b66=0
b65=0
b64=(6*a22*b2)/a2
b63=0
b62=(a22*b2)/a2
b61=(6*a22*b2)/a3
b60=(18*a22*b2)/a3
b59=0
b58=0
b57=0
b56=0
b55=( - 1/486*a3**5*b2)/a2**5
b54=0
b53=( - 5/81*a3**4*b2)/a2**4
b52=0
b51=0
b50=( - 5/18*a3**3*b2)/a2**3
b49=( - 5/27*a3**3*b2)/a2**3
b48=0
b47=0
b46=0
b45=0
b44=( - 5/9*a3**2*b2)/a2**2
b43=( - 5/18*a3**2*b2)/a2**2
b42=0
b41=0
b40=0
b39=0
b38=( - 1/6*a3*b2)/a2
b37=0
b36=0
b35=0
b34=0
b33=0
b32=0
b31=0
b30=0
b29=0
b28=( - 4/3*a22*a3*b2)/a2**2
b27=( - 2*a22*a3*b2)/a2**2
b26=( - 12*a22*b2)/a2
b25=( - 12*a22*b2)/a2
b24=( - 36*a22*b2)/a2
b23=( - 6*a22*b2)/a2
b22=( - 6*a22*b2)/a2
b21=( - 36*a22*b2)/a3
b20=( - 108*a22*b2)/a3
b19=( - 72*a22*b2)/a3
b18=( - 72*a22*b2)/a3
b17=( - 48*a22*b2)/a3
b16=( - 6*a22*b2)/a3
b15=( - 36*a2*a22*b2)/a3**2
b14=( - 144*a2*a22*b2)/a3**2
b13=( - 108*a2*a22*b2)/a3**2
b12=0
b11=(5/81*a3**4*b2)/a2**4
b10=(10/9*a3**3*b2)/a2**3
b9=(10/27*a3**3*b2)/a2**3
b8=(5/3*a3**2*b2)/a2**2
b7=(50/9*a3**2*b2)/a2**2
b6=(10/9*a3**2*b2)/a2**2
b5=(10/3*a3*b2)/a2
b4=(5*a3*b2)/a2
b3=(5/3*a3*b2)/a2
a21=(6*a2*a22)/a3
a20=0
a19=( - 1/54*a3**3)/a2**2
a18=0
a17=( - 1/6*a3**2)/a2
a16=0
a15=0
a14= - 1/6*a3
a13=0
a12=0
a11=0
a10=( - 12*a2*a22)/a3
a9=( - 6*a2*a22)/a3
a8=( - 36*a2**2*a22)/a3**2
a7=( - 36*a2**2*a22)/a3**2
a6=0
a5=(1/3*a3**2)/a2
a4=a3
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:
b2,a22,a3,a2
Relevance for the application:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\$
2 2 3 2 3 2
df(u,t)=(u *a2 *a3 + u *a2*a3 *u + u *a2*a3 - 6*u *f(v,v)*a2 *a22*a3
3x 2x x x
1 4 2 3 3
+ ---*u *a3 *u - 36*f(v ,v )*a2 *a22 - 36*f(v,v )*a2 *a22
3 x x x 2x
2 2
- 12*f(v,v )*a2 *a22*a3*u)/(a2*a3 )
x
1 2 2 1 3 3
df(v,t)=( - ---*u *a2 *a3 *v - ---*u *a2*a3 *u*v + 6*f(v,v )*a2 *a22*v
6 2x 6 x x
2 1 4 3 2
+ f(v,v)*a2 *a22*a3*u*v - ----*a3 *u *v)/(a2 *a3)
54
4 2 5 3 3 3 3
df(u,s)=(u *a2 *a3 *b2 + ---*u *a2 *a3 *b2*u + 5*u *u *a2 *a3 *b2
5x 3 4x 3x x
4 10 2 4 2
- 6*u *f(v,v)*a2 *a22*a3*b2 + ----*u *a2 *a3 *b2*u
3x 9 3x
10 2 3 3 50 2 4
+ ----*u *a2 *a3 *b2 + ----*u *u *a2 *a3 *b2*u
3 2x 9 2x x
4 3 2
- 48*u *f(v,v )*a2 *a22*a3*b2 - 6*u *f(v,v)*a2 *a22*a3 *b2*u
2x x 2x
10 5 3 5 3 2 4
+ ----*u *a2*a3 *b2*u + ---*u *a2 *a3 *b2
27 2x 3 x
2 3 2 10 2 5 2
- 6*u *f(v,v)*a2 *a22*a3 *b2 + ----*u *a2*a3 *b2*u
x 9 x
4 4
- 72*u *f(v ,v )*a2 *a22*a3*b2 - 72*u *f(v,v )*a2 *a22*a3*b2
x x x x 2x
3 2 2 3 2
- 36*u *f(v,v )*a2 *a22*a3 *b2*u - 2*u *f(v,v)*a2 *a22*a3 *b2*u
x x x
5 6 4 5
+ ----*u *a3 *b2*u - 108*f(v ,v )*a2 *a22*b2
81 x 2x 2x
5 4
- 144*f(v ,v )*a2 *a22*b2 - 108*f(v ,v )*a2 *a22*a3*b2*u
x 3x x 2x
3 2 2 5
- 12*f(v ,v )*a2 *a22*a3 *b2*u - 36*f(v,v )*a2 *a22*b2
x x 4x
4 3 2 2
- 36*f(v,v )*a2 *a22*a3*b2*u - 12*f(v,v )*a2 *a22*a3 *b2*u
3x 2x
4 2 3 3 4 2
- ---*f(v,v )*a2 *a22*a3 *b2*u )/(a2 *a3 )
3 x
1 4 2 5 3 3
df(v,s)=( - ---*u *a2 *a3 *b2*v - ----*u *a2 *a3 *b2*u*v
6 4x 18 3x
5 3 3 4
- ---*u *u *a2 *a3 *b2*v + u *f(v,v)*a2 *a22*a3*b2*v
9 2x x 2x
5 2 4 2 5 2 2 4
- ----*u *a2 *a3 *b2*u *v - ----*u *a2 *a3 *b2*u*v
27 2x 18 x
4 3 2
+ 6*u *f(v,v )*a2 *a22*a3*b2*v + u *f(v,v)*a2 *a22*a3 *b2*u*v
x x x
5 5 3 5
- ----*u *a2*a3 *b2*u *v + 18*f(v ,v )*a2 *a22*b2*v
81 x x 2x
4 5
+ 6*f(v ,v )*a2 *a22*a3*b2*u*v + 6*f(v,v )*a2 *a22*b2*v
x x 3x
4 3 2 2
+ 6*f(v,v )*a2 *a22*a3*b2*u*v + 2*f(v,v )*a2 *a22*a3 *b2*u *v
2x x
1 2 3 3 1 6 5 5
+ ---*f(v,v)*a2 *a22*a3 *b2*u *v - -----*a3 *b2*u *v)/(a2 *a3)
9 486
df(u,t)=(df(u,x,3)*a2**2*a3**2 + df(u,x,2)*a2*a3**3*u + df(u,x)**2*a2*a3**3 - 6*
df(u,x)*f(v,v)*a2**2*a22*a3 + 1/3*df(u,x)*a3**4*u**2 - 36*f(df(v,x),df(v,x))*a2
**3*a22 - 36*f(v,df(v,x,2))*a2**3*a22 - 12*f(v,df(v,x))*a2**2*a22*a3*u)/(a2*a3**
2)$
df(v,t)=( - 1/6*df(u,x,2)*a2**2*a3**2*v - 1/6*df(u,x)*a2*a3**3*u*v + 6*f(v,df(v,
x))*a2**3*a22*v + f(v,v)*a2**2*a22*a3*u*v - 1/54*a3**4*u**3*v)/(a2**2*a3)$
df(u,s)=(df(u,x,5)*a2**4*a3**2*b2 + 5/3*df(u,x,4)*a2**3*a3**3*b2*u + 5*df(u,x,3)
*df(u,x)*a2**3*a3**3*b2 - 6*df(u,x,3)*f(v,v)*a2**4*a22*a3*b2 + 10/9*df(u,x,3)*a2
**2*a3**4*b2*u**2 + 10/3*df(u,x,2)**2*a2**3*a3**3*b2 + 50/9*df(u,x,2)*df(u,x)*a2
**2*a3**4*b2*u - 48*df(u,x,2)*f(v,df(v,x))*a2**4*a22*a3*b2 - 6*df(u,x,2)*f(v,v)*
a2**3*a22*a3**2*b2*u + 10/27*df(u,x,2)*a2*a3**5*b2*u**3 + 5/3*df(u,x)**3*a2**2*
a3**4*b2 - 6*df(u,x)**2*f(v,v)*a2**3*a22*a3**2*b2 + 10/9*df(u,x)**2*a2*a3**5*b2*
u**2 - 72*df(u,x)*f(df(v,x),df(v,x))*a2**4*a22*a3*b2 - 72*df(u,x)*f(v,df(v,x,2))
*a2**4*a22*a3*b2 - 36*df(u,x)*f(v,df(v,x))*a2**3*a22*a3**2*b2*u - 2*df(u,x)*f(v,
v)*a2**2*a22*a3**3*b2*u**2 + 5/81*df(u,x)*a3**6*b2*u**4 - 108*f(df(v,x,2),df(v,x
,2))*a2**5*a22*b2 - 144*f(df(v,x),df(v,x,3))*a2**5*a22*b2 - 108*f(df(v,x),df(v,x
,2))*a2**4*a22*a3*b2*u - 12*f(df(v,x),df(v,x))*a2**3*a22*a3**2*b2*u**2 - 36*f(v,
df(v,x,4))*a2**5*a22*b2 - 36*f(v,df(v,x,3))*a2**4*a22*a3*b2*u - 12*f(v,df(v,x,2)
)*a2**3*a22*a3**2*b2*u**2 - 4/3*f(v,df(v,x))*a2**2*a22*a3**3*b2*u**3)/(a2**4*a3
**2)$
df(v,s)=( - 1/6*df(u,x,4)*a2**4*a3**2*b2*v - 5/18*df(u,x,3)*a2**3*a3**3*b2*u*v -
5/9*df(u,x,2)*df(u,x)*a2**3*a3**3*b2*v + df(u,x,2)*f(v,v)*a2**4*a22*a3*b2*v - 5
/27*df(u,x,2)*a2**2*a3**4*b2*u**2*v - 5/18*df(u,x)**2*a2**2*a3**4*b2*u*v + 6*df(
u,x)*f(v,df(v,x))*a2**4*a22*a3*b2*v + df(u,x)*f(v,v)*a2**3*a22*a3**2*b2*u*v - 5/
81*df(u,x)*a2*a3**5*b2*u**3*v + 18*f(df(v,x),df(v,x,2))*a2**5*a22*b2*v + 6*f(df(
v,x),df(v,x))*a2**4*a22*a3*b2*u*v + 6*f(v,df(v,x,3))*a2**5*a22*b2*v + 6*f(v,df(v
,x,2))*a2**4*a22*a3*b2*u*v + 2*f(v,df(v,x))*a2**3*a22*a3**2*b2*u**2*v + 1/9*f(v,
v)*a2**2*a22*a3**3*b2*u**3*v - 1/486*a3**6*b2*u**5*v)/(a2**5*a3)$