@mehdi jafari So Xstar is a function of T. Thus the first equation should be written maybe as
EQ[1]:=(1/Dg)*diff(X(T,Z),T)+diff(X(T,Z),Z)+3*St*(X(T,Z)-Y(T,1)^(1/n));

I don't see any way to handle this equation together with EQ[2].

@Carl Love I agree. As I said in my answer: If you start at (theta,theta_p) =(Pi,0) you will never leave, if you don't, you will never get there!
So Kitonum's remark in his answer: "At this value there is instability when the pendulum reaches the top position." is misleading. The pendulum cannot reach the top position and also be at rest!

@mskalsi I agree that frontend is not easy to understand, so I appreciate the freeze/thaw approach used by Carl.

Incidentally, the first term in your input expression had the factor diff(g(z),x), where surely you meant diff(g(z),z).

To give an example, where more freezing is necessary (and by default handled by frontend) take this modified expression:

eq2 := sin(g(z))*diff(g(z), z)*g(z)^3+diff(g(z), z, z)*g(z)^4+diff(g(z), z, z, z)*g(z)^5+diff(g(z), z)/g(z)^2;

The command
thaw(coeff(subsindets[flat](eq2, specfunc(diff), freeze), g(z),3));
#doesn't succeed because sin(g(z)) won't be frozen and because coeff doesn't work here:
coeff(sin(g(z))*F*g(z)^3,g(z),3);
#By default frontend also freezes sin(g(z)) as is illustrated here
frontend(diff,[sin(g(z)),z]); #returns zero
frontend(diff,[sin(g(z)),z],[{`+`,`*`,function},{}]); #Doesn't freeze expressions of type function.
#So frontend succeeds in
frontend(coeff,[eq2,g(z),3]);
# but not in
frontend(coeff,[eq2,g(z),3],[{`+`,`*`,function},{}]);

@Kitonum It appears that it is the cook method which gives erroneous results:
restart;
res:=[seq(int(1/(x^4+p), x=0..1,method=_RETURNVERBOSE), p=1/2..5, 1/2)];
evalindets(res,`=`(identical(FAILS),anything),x->NULL);
evalf(%);
restart;
res:=[seq(int(1/(x^4+p), x=0..1,method=cook), p=1/2..5, 1/2)];
###################################
I tried for general p with and without the assumption p>0.

## Without assumptions:
restart;
infolevel[int]:=3:
res:=int(1/(x^4+p), x=0..1);
simplify(evalc(eval(res,p=4))); #cook correct
evalf(%);
##With the assumption p>0:
restart;
infolevel[int]:=3:
res:=int(1/(x^4+p), x=0..1) assuming p>0;
simplify(eval(res,p=4)); #cook wrong

## Just in case this bug has not been reported as an SCR, I shall do it.

@torabi  Just a comment: Your latest system is linear, so there is no particularly good reason to use small values of b like b=10^(-5). You might as well use b=1.

@ThU Indeed, Maple 17 works on my 64 bit computer with Windows 10 Home Edition.

@Carl Love To put the system into a form that allows you to solve for the highest derivatives you can differentiate the first of the equations. This won't introduce any higher derivatives in the system:

ode1:=diff(sys[1],x);
solve({sys[2],ode1}, {diff(w(x),x$4), diff(psi(x),x$3)}); # Possible
# Now add a boundary condition to ensure that the new system is equivalent to the old:
bcs1:=eval(convert(sys[1],D),x=1); # Might as well have used x = 0.
# So new boundary conditions are
BCS:=bcs union {bcs1};
nops(BCS);
## To these must be added the extra condition to determine omega2.
##
By doing that I get 8 successes out of 9 tries without any optional arguments to dsolve expect for the output option, i.e. only
dsolve({sys[2],ode1} union BCS union {b = 10^(-5)}, numeric,output=listprocedure) . 

'undefined' is not the result of an error.
Example:
Heaviside(0);
                                  undefined
But 0/0 produces the error
                            Error, numeric exception: division by zero
i.e. it doesn't result in 'undefined'.

I suppose that you have already looked at the help page:

?undefined

@mwahab I don't know much about this (actually close to nothing), but I tried the following where my result RES differs from the result obtained by map(pdsolve, [res2], parameters = {m}) in the latter missing the case m = 0.

restart;
with(PDEtools);
declare(u(t, x))
pde:=diff(u(t, x), t)-u(t,x)^m*(diff(u(t, x), x))-u(t,x)^m-u(t,x)^m*(diff(u(t, x), x, x))-u(t,x)^m*(diff(u(t, x), x, x, x)) = 0;
res:=DeterminingPDE(pde, u(t, x), integrabilityconditions = false);
nops(res);
res2:=casesplit(res, parameters = {m});
map(pdsolve, [res2], parameters = {m});
n:=nops([res2]);
map2(op,2,[res2]); # The &where's
eqs:=map2(op,1,[res2]); #The equations including conditions on m
sys:=map2(select,has,eqs,{_eta,_xi}); #the pdes
M:=map2(remove,has,eqs,{_eta,_xi}); # the m equations
M1:=subs([]=[m=m,m=m],M); #Slightly artificial
M2:=map2(op,2,M1); #Final version of m equations
RES:=seq( [M2[i],pdsolve(eval(sys[i],M2[i]))],i=1..n); #result
map(pdsolve, [res2], parameters = {m});

See the help page for dsolve:
?dsolve

Include any initial conditions to determine the arbitrary constant(s).

Have a look at the help page
?dsolve,numeric

It has examples of ivp as well as bvp problems.

Note: I must agree with vv that this seems to be a case of using a generic formula.
So I converted this into a comment.
Here is a simplified version, where I set t = 0 and a=b=2 right away.

restart;
G:=n->sum(cos(2*Pi/n*j)^2*sin(2*Pi/n*j)^2,j=0..n-1); #The simplified version
Ga:=n->add(cos(2*Pi/n*j)^2*sin(2*Pi/n*j)^2,j=0..n-1); #Version using add
H:=n->sum(combine(cos(2*Pi/n*j)^2*sin(2*Pi/n*j)^2),j=0..n-1); #Version combining before summing
G(n); #Wrong for some n
H(n); #Seems to be better
combine~([seq(G(n),n=1..5)]);
combine~([seq(Ga(n),n=1..5)]);
[seq(H(n),n=1..5)];

##Test of n/8 as the answer:
simplify(combine~([seq(G(n)-n/8,n=1..25)]));
simplify(combine~([seq(Ga(n)-n/8,n=1..25)]));
simplify([seq(H(n)-n/8,n=1..25)]);

@Carl Love His expected output can be obtained by doing

PDEtools:-declare(phi(X));
Diff(conjugate(phi(X)),x1);
                             

I checked that mgear is still present in Maple 7 by doing
interface(verboseproc=2);
eval(`dsolve/numeric/mgear`);

I looked at the help page for dsolve/numeric/mgear in Maple 7 via What's New. There it says:

"The mgear method is obsolete, and is no longer available in Maple. The rosenbrock and lsode methods are avaialble for the numeric solutions of stiff initial value problems."

@olivertwist My code works in Maple 2016 (but it doesn't use Douglas Meade's Shoot package, so you don't need the with(Shoot) line).

Which Maple version do you have?
The eval[recurse] command is only available in more recent versions. If the result of that command (i.e. sys) is not free of variables with p added to them (as in fp) then you must have an older version.
In older versions you can do the following (which also works in more recent versions):

SBS:=fppp(eta) = diff(fpp(eta), eta), fpp(eta) = diff(fp(eta), eta), fp(eta) = diff(f(eta), eta),gp(eta) = diff(g(eta), eta), mp(eta) = diff(m(eta), eta), np(eta) = diff(n(eta), eta); #The order is important!!!
sys:=subs(SBS,sys1); #The first substitution is applied first, etc.

#####
I'm puzzled by your remark "I solved the problem using desolve without converting the system into first order one (RKF45)".
If you used dsolve with boundary conditions given at two points (here eta=0, eta=blt=1) then you were not using RKF45 as that method is for initial value problems only.