This part
(diff(v(r,t),r)-v(r,t)/r+beta*diff(diff(v(r,t),r),t)
- 1/r * beta*diff(v(r,t),t)
-Re*(1+lambda)*exp(1-p(r,t))*diff(v(r,t),t)-(1+lambda)*exp(1-p(r,t))*g2
starts with a parenthesis '(' , but its partner ')' is missing.
Furthermore, Re has a multiplication sign following it. It should just be Re(....).
Your IBC has sint, it should be sin(t).

@wzelik I get an error 404. File or directory not found.

@Carl Love My guess is that the intended meaning was plotting z as a function of x which is accomplished as Adri van der Meer is describing or as I commented to the asker's question.
But only the asker can tell us, what he meant. Clearly these different interpretations of the intended meaning are quite different.

It is not clear to me what you want to do: What does "after indicate to extract u(t) etc. " mean?

Change the last line to
odeplot(sol,[x(t),z(t)],0..10,numpoints=200,color=blue,legend="z",scaling=constrained);

@bunmipopoola From the picture you posted it appears that the maximal time considered is 3.1 ms, i.e. 3.1*10^(-3) s. Why do you use .0000031? Is the unit of time not seconds?
Are you sure that you have the correct equation eq1? Is the sign simply wrong?
It follows from eq1 that D(F)(t) > 0 for all time t. Thus F is increasing. If F is the amount of fuel, then that is clearly wrong, since you are not gathering wood for your wood burning stove.

@bunmipopoola Did you try all my lines?
Try copying the whole thing from my answer and paste it into a fresh worksheet. Then execute.
I just tried that very thing myself in Maple 17 (but the version shouldn't matter much).
No problem.

@bunmipopoola Did you try all my lines?
Try copying the whole thing from my answer and paste it into a fresh worksheet. Then execute.
I just tried that very thing myself in Maple 17 (but the version shouldn't matter much).
No problem.

@Gaia Now the issue with +- seems  resolved: It meant -.
You need eqt to be a polynomial not an equation. Simply remove =0.
You won't have any luck with algcurves[parametrize] though.

@traiannp If the system really is inconsistent, then that means there just is no solution!

@traiannp If the system really is inconsistent, then that means there just is no solution!

In Maple 17
solve(sys, [c,d,e,f,g,h,i,j,k,l,m,n,p]);

returns [], i.e. it didn't find any solution.

In Maple 17
solve(sys, [c,d,e,f,g,h,i,j,k,l,m,n,p]);

returns [], i.e. it didn't find any solution.

This simpler version also returns an incorrect result:

s:=HFloat(-infinity);
type(s,positive);
returns true.

Could you correct the syntax of your input
eqt := y^5+y^4 + -0.5109*y^3 + -0.0595*y^2 + 0.0000-(-3.1086e-015*u^4 + u^3 + -0.5109*u^2 + -0.0595*u) = 0;
so it parses in Maple?
It is not clear to me what you intended it to mean, so I won't try myself.