Question:Problem with assumptions and pdsolve

Question:Problem with assumptions and pdsolve

Maple 2019

I have some trouble solving the pde of:
ut + u^2*ux = u, u(x,0) = x, with x::real and t>0.

I think that I have 2 problems.

1.
The first part of the code I define u(x,t) with both the variable rp.
Then I define the variable q copying the definition of u(x,t).
When I try to insert q and u(x,t) in the initial equation - one is able to be reduced to one term while the other isn't.
So I'm not really sure what is happening here.

2.
When I use Maple's pdsolve() I get a result, but when I insert the answer in the initial equation - then it isn't correct.
I tried to show this in the last part of the code.

 > # ut + u^2*ux = u, u(x,0) = x
 > restart
 > rp := (-1 + sqrt(1 + 4*exp(t)^2*t*x))/(2*exp(t)^2*t);
 (1)
 > u := (x,t) -> rp*exp(t): 'u(x,t)' = u(x,t);
 (2)
 > q := (-1 + sqrt(1 + 4*exp(t)^2*t*x))/(2*exp(t)*t); # Copying the result from above and defining q the same
 (3)
 > # Now doing the same operations on supposedly the same term, but one is able to be reduced with assumptions while the other isn't.
 > L_nothing := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) ; L_real := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) assuming x::real; L_t := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) assuming t>0; L_all := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) assuming t>0, x::real;
 (4)
 > L_nothing := diff(q,t) + q^2*diff(q,x) ; L_real := diff(q,t) + q^2*diff(q,x) assuming x::real; L_t := diff(q,t) + q^2*diff(q,x) assuming t>0; L_all := diff(q,t) + q^2*diff(q,x) assuming t>0, x::real;
 (5)
 > restart
 > pde := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) = u(x,t); ic := u(x,0) = x;
 (6)
 > pdsolve([pde, ic]);
 (7)
 > u := (x,t) -> exp(t)*(sqrt(2*exp(2*t)*x - 2*x + 1) - 1)/(exp(2*t) - 1);
 (8)
 > L := diff(u(x,t),t) + u(x,t)^2*diff(u(x,t),x) assuming t>0, x::real;
 (9)
 > LL := simplify(L) = u(x,t)
 (10)
 > evalb(LL)
 (11)
 > # Obviously not correct solution... or what?
 >