restart;
P := -lambda*exp(-Phi(xi))-mu*exp(Phi(xi));
-lambda exp(-Phi(xi)) - mu exp(Phi(xi))
u[0] := A[0]+A[1]*exp(-Phi(xi))+A[2]*exp(-Phi(xi))*exp(-Phi(xi));
2
A[0] + A[1] exp(-Phi(xi)) + A[2] (exp(-Phi(xi)))
u[1] := diff(u[0], xi);
/ d \
-A[1] |---- Phi(xi)| exp(-Phi(xi))
\ dxi /
2 / d \
- 2 A[2] (exp(-Phi(xi))) |---- Phi(xi)|
\ dxi /
d[1] := -A[1]*P*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*P;
-A[1] (-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) exp(-Phi(xi)) - 2
2
A[2] (exp(-Phi(xi))) (-lambda exp(-Phi(xi)) - mu exp(Phi(xi)))
u[2] := diff(d[1], xi);
/ / d \
-A[1] |lambda |---- Phi(xi)| exp(-Phi(xi))
\ \ dxi /
/ d \ \
- mu |---- Phi(xi)| exp(Phi(xi))| exp(-Phi(xi)) + A[1] (
\ dxi / /
/ d \
-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) |---- Phi(xi)| exp(-Phi(
\ dxi /
2
xi)) + 4 A[2] (exp(-Phi(xi))) (-lambda exp(-Phi(xi))
/ d \ 2
- mu exp(Phi(xi))) |---- Phi(xi)| - 2 A[2] (exp(-Phi(xi)))
\ dxi /
/ / d \
|lambda |---- Phi(xi)| exp(-Phi(xi))
\ \ dxi /
/ d \ \
- mu |---- Phi(xi)| exp(Phi(xi))|
\ dxi / /
d[2] := -A[1]*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)))*exp(-Phi(xi))+A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P*exp(-Phi(xi))+4*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P-2*A[2]*(exp(-Phi(xi)))^2*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)));
-A[1] (lambda (-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) exp(-Phi(
xi))
- mu (-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) exp(Phi(xi)))
exp(-Phi(xi))
2
+ A[1] (-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) exp(-Phi(xi)) + 4
2
A[2] (exp(-Phi(xi)))
2
(-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) - 2 A[2]
2
(exp(-Phi(xi))) (lambda (-lambda exp(-Phi(xi))
- mu exp(Phi(xi))) exp(-Phi(xi))
- mu (-lambda exp(-Phi(xi)) - mu exp(Phi(xi))) exp(Phi(xi)))
collect(expand((2*k*k)*w*beta*d[2]-(2*alpha*k*k)*d[1]-2*w*u[0]+k*u[0]*u[0]), exp(Phi(xi)));
2 2 2 2
4 k w beta A[2] mu - 2 alpha k A[1] mu + k A[0] - 2 w A[0] +
1 / 2 2
------------ \4 beta k lambda mu w A[1] - 4 alpha k mu A[2]
exp(Phi(xi))
\ 1 / 2
+ 2 k A[0] A[1] - 2 w A[1]/ + --------------- \16 beta k
2
(exp(Phi(xi)))
2
lambda mu w A[2] - 2 alpha k lambda A[1] + 2 k A[0] A[2]
2 \ 1 / 2 2
+ k A[1] - 2 w A[2]/ + --------------- \4 beta k lambda w A
3
(exp(Phi(xi)))
2 \
[1] - 4 alpha k lambda A[2] + 2 k A[1] A[2]/
2 2 2
12 beta k lambda w A[2] + k A[2]
+ -----------------------------------
4
(exp(Phi(xi)))
restart;
solve({12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2, 4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2], 4*beta*k^2*mu^2*w*A[2]-2*alpha*k^2*mu*A[1]+k*A[0]^2-2*w*A[0], 4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1], 16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2]}, {k, w, A[0], A[1], A[2]});
{k = 0, w = 0, A[0] = A[0], A[1] = A[1], A[2] = A[2]},
{k = k, w = w, A[0] = 0, A[1] = 0, A[2] = 0},
/
/ 2 w \ |
{ k = k, w = w, A[0] = ---, A[1] = 0, A[2] = 0 }, <
\ k / |
\
/ 2 \
k = RootOf\24 beta lambda mu _Z - 1/,
/ 2 \
RootOf\100 lambda mu _Z + 1/ alpha
w = -----------------------------------,
beta
/ 2 \
RootOf\100 lambda mu _Z + 1/ alpha
A[0] = ----------------------------------------,
/ 2 \
2 beta RootOf\24 beta lambda mu _Z - 1/
alpha
A[1] = --------------------------------------------, A[2] = -12
/ 2 \
10 beta mu RootOf\24 beta lambda mu _Z - 1/
/ 2 \ 2 /
RootOf\24 beta lambda mu _Z - 1/ lambda RootOf\100 lambda mu
\ /
2 \ | | / 2 \
_Z + 1/ alpha >, < k = RootOf\24 beta lambda mu _Z + 1/,
| |
/ \
/ 2 \
RootOf\100 lambda mu _Z + 1/ alpha
w = -----------------------------------,
beta
/ 2 \
3 RootOf\100 lambda mu _Z + 1/ alpha
A[0] = ----------------------------------------,
/ 2 \
2 beta RootOf\24 beta lambda mu _Z + 1/
alpha
A[1] = - --------------------------------------------, A[2] = -
/ 2 \
10 beta mu RootOf\24 beta lambda mu _Z + 1/
/ 2 \ 2 /
12 RootOf\24 beta lambda mu _Z + 1/ lambda RootOf\100 lambda
\
2 \ |
mu _Z + 1/ alpha >
|
/
set 1;
Error, missing operation
Typesetting:-mambiguous(Typesetting:-mambiguous(set 1,
Typesetting:-merror("missing operation")))
{k = RootOf(24*_Z^2*beta*lambda*mu-1), w = RootOf(100*_Z^2*lambda*mu+1)*alpha/beta, A[0] = (1/2)*RootOf(100*_Z^2*lambda*mu+1)*alpha/(beta*RootOf(24*_Z^2*beta*lambda*mu-1)), A[1] = (1/10)*alpha/(beta*mu*RootOf(24*_Z^2*beta*lambda*mu-1)), A[2] = -12*RootOf(24*_Z^2*beta*lambda*mu-1)*lambda^2*RootOf(100*_Z^2*lambda*mu+1)*alpha};
/
| / 2 \
< k = RootOf\24 beta lambda mu _Z - 1/,
|
\
/ 2 \
RootOf\100 lambda mu _Z + 1/ alpha
w = -----------------------------------,
beta
/ 2 \
RootOf\100 lambda mu _Z + 1/ alpha
A[0] = ----------------------------------------,
/ 2 \
2 beta RootOf\24 beta lambda mu _Z - 1/
alpha
A[1] = --------------------------------------------, A[2] = -12
/ 2 \
10 beta mu RootOf\24 beta lambda mu _Z - 1/
/ 2 \ 2 /
RootOf\24 beta lambda mu _Z - 1/ lambda RootOf\100 lambda mu
\
2 \ |
_Z + 1/ alpha >
|
/
restart;
restart;
solve({24*Z^2*beta*lambda*mu-1}, {Z});
/ (1/2) \
| 6 |
< Z = ------------------------ >,
| (1/2)|
\ 12 (beta lambda mu) /
/ (1/2) \
| 6 |
< Z = - ------------------------ >
| (1/2)|
\ 12 (beta lambda mu) /
solve({100*Z^2*lambda*mu+1}, {Z});
/ 1 \ / 1 \
|Z = - --------------------| , |Z = --------------------|
< (1/2) > < (1/2) >
| 10 (-lambda mu) | | 10 (-lambda mu) |
\ / \ /
restart;
k := (1/12)*sqrt(6)/sqrt(beta*lambda*mu);
(1/2)
6
------------------------
(1/2)
12 (beta lambda mu)
w := -alpha/((10*sqrt(-lambda*mu))*beta);
alpha
- -------------------------
(1/2)
10 (-lambda mu) beta
A[0] := 1/2*(-alpha/((10*sqrt(-lambda*mu))*((1/12)*beta*sqrt(6)/sqrt(beta*lambda*mu))));
(1/2) (1/2)
alpha 6 (beta lambda mu)
- ----------------------------------
(1/2)
10 (-lambda mu) beta
A[1] := (1/10)*alpha/((1/12)*beta*mu*sqrt(6)/sqrt(beta*lambda*mu));
(1/2) (1/2)
alpha 6 (beta lambda mu)
----------------------------------
5 beta mu
A[2] := (12*(1/12))*sqrt(6)*lambda^2*alpha/(sqrt(beta*lambda*mu)*(10*sqrt(-lambda*mu)));
(1/2) 2
6 lambda alpha
------------------------------------------
(1/2) (1/2)
10 (beta lambda mu) (-lambda mu)
lambda := 3;
3
mu := 2;
2
H := -ln(sqrt(lambda/mu)*tan(sqrt(lambda*mu)*(xi+C)));
/1 (1/2) / (1/2) \\
-ln|- 6 tan\6 (xi + C)/|
\2 /
u[0] := A[0]+A[1]*exp(-H)+A[2]*exp(-H)*exp(-H);
(1/2) (1/2) / (1/2) \
alpha (-6) 3 alpha 6 tan\6 (xi + C)/
--------------- + -----------------------------------
(1/2) (1/2)
10 beta 10 beta
2
(1/2) / (1/2) \
9 alpha (-6) tan\6 (xi + C)/
- ---------------------------------------
(1/2)
40 beta
f := diff(u[0], xi);
/ 2\
| / (1/2) \ | /
9 alpha \1 + tan\6 (xi + C)/ / 1 |
----------------------------------- - ------------ \9 alpha
(1/2) (1/2)
5 beta 20 beta
/ 2\
(1/2) / (1/2) \ | / (1/2) \ |
(-6) tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ /
\
(1/2)|
6 /
S := diff(f, xi);
/ 2\
/ (1/2) \ | / (1/2) \ | (1/2)
18 alpha tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ / 6
----------------------------------------------------------------
(1/2)
5 beta
2
/ 2\
(1/2) | / (1/2) \ |
27 alpha (-6) \1 + tan\6 (xi + C)/ /
- ----------------------------------------------- -
(1/2)
10 beta
/ 2 /
1 | (1/2) / (1/2) \ |
----------- \27 alpha (-6) tan\6 (xi + C)/ \1
(1/2)
5 beta
2\\
/ (1/2) \ ||
+ tan\6 (xi + C)/ //
eq := (2*k*k)*w*beta*S-(2*alpha*k*k)*f-2*w*u[0]+k*u[0]*u[0];
/ /
| |
| |
1 | (1/2) |
--------- |alpha (-6) |
4320 beta | |
\ \
/ 2\
/ (1/2) \ | / (1/2) \ | (1/2)
18 alpha tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ / 6
----------------------------------------------------------------
(1/2)
5 beta
2
/ 2\
(1/2) | / (1/2) \ |
27 alpha (-6) \1 + tan\6 (xi + C)/ /
- ----------------------------------------------- -
(1/2)
10 beta
/ 2 /
1 | (1/2) / (1/2) \ |
----------- \27 alpha (-6) tan\6 (xi + C)/ \1
(1/2)
5 beta
\\
|| / /
2\\|| | |
/ (1/2) \ |||| 1 | |
+ tan\6 (xi + C)/ //|| - ------- |alpha |
|| 72 beta | |
// \ \
/ 2\
| / (1/2) \ | /
9 alpha \1 + tan\6 (xi + C)/ / 1 |
----------------------------------- - ------------ \9 alpha
(1/2) (1/2)
5 beta 20 beta
/ 2\
(1/2) / (1/2) \ | / (1/2) \ |
(-6) tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ /
\\ / /
\|| | | (1/2)
(1/2)||| 1 | (1/2) |alpha (-6)
6 /|| - ------- |alpha (-6) |---------------
|| 30 beta | | (1/2)
// \ \ 10 beta
(1/2) / (1/2) \
3 alpha 6 tan\6 (xi + C)/
+ -----------------------------------
(1/2)
10 beta
2\\ /
(1/2) / (1/2) \ || |
9 alpha (-6) tan\6 (xi + C)/ || 1 |
- ---------------------------------------|| + ------------ |
(1/2) || (1/2) |
40 beta // 12 beta \
/
| (1/2) (1/2) / (1/2) \
|alpha (-6) 3 alpha 6 tan\6 (xi + C)/
|--------------- + -----------------------------------
| (1/2) (1/2)
\ 10 beta 10 beta
2\ \
(1/2) / (1/2) \ | |
9 alpha (-6) tan\6 (xi + C)/ | |
- ---------------------------------------|^2|
(1/2) | |
40 beta / /
value(%);
/ /
| |
| |
1 | (1/2) |
--------- |alpha (-6) |
4320 beta | |
\ \
/ 2\
/ (1/2) \ | / (1/2) \ | (1/2)
18 alpha tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ / 6
----------------------------------------------------------------
(1/2)
5 beta
2
/ 2\
(1/2) | / (1/2) \ |
27 alpha (-6) \1 + tan\6 (xi + C)/ /
- ----------------------------------------------- -
(1/2)
10 beta
/ 2 /
1 | (1/2) / (1/2) \ |
----------- \27 alpha (-6) tan\6 (xi + C)/ \1
(1/2)
5 beta
\\
|| / /
2\\|| | |
/ (1/2) \ |||| 1 | |
+ tan\6 (xi + C)/ //|| - ------- |alpha |
|| 72 beta | |
// \ \
/ 2\
| / (1/2) \ | /
9 alpha \1 + tan\6 (xi + C)/ / 1 |
----------------------------------- - ------------ \9 alpha
(1/2) (1/2)
5 beta 20 beta
/ 2\
(1/2) / (1/2) \ | / (1/2) \ |
(-6) tan\6 (xi + C)/ \1 + tan\6 (xi + C)/ /
\\ / /
\|| | | (1/2)
(1/2)||| 1 | (1/2) |alpha (-6)
6 /|| - ------- |alpha (-6) |---------------
|| 30 beta | | (1/2)
// \ \ 10 beta
(1/2) / (1/2) \
3 alpha 6 tan\6 (xi + C)/
+ -----------------------------------
(1/2)
10 beta
2\\ /
(1/2) / (1/2) \ || |
9 alpha (-6) tan\6 (xi + C)/ || 1 |
- ---------------------------------------|| + ------------ |
(1/2) || (1/2) |
40 beta // 12 beta \
/
| (1/2) (1/2) / (1/2) \
|alpha (-6) 3 alpha 6 tan\6 (xi + C)/
|--------------- + -----------------------------------
| (1/2) (1/2)
\ 10 beta 10 beta
2\ \
(1/2) / (1/2) \ | |
9 alpha (-6) tan\6 (xi + C)/ | |
- ---------------------------------------|^2|
(1/2) | |
40 beta / /
simplify(%);
/ /
1 | 2 | / (1/2)
----------------------------------- \alpha \24 I sin\6 (xi
4
(3/2) / (1/2) \
640 beta cos\6 (xi + C)/
3 4
\ / (1/2) \ / (1/2) \
+ C)/ cos\6 (xi + C)/ - 21 cos\6 (xi + C)/
/ (1/2) \ / (1/2) \
- 16 I sin\6 (xi + C)/ cos\6 (xi + C)/
2 \\
/ (1/2) \ ||
+ 26 cos\6 (xi + C)/ - 9//
