Confused about the terminology for the domain pde

[oddiseas]

Homework Statement

δu/δt+2tδu/δx=1

for t>0,x>0 with u= 0 on x= 0 for t>0, u=1 at t=0 for x≥0

Homework Equations

The Attempt at a Solution

((dx)/(dt))=2t
x=t²+c
x-t²=c
the general solution is:
u=t+F(x-t²)
Now i am confused about the terminology for the domain, ie it says it is defined for t>0, x>0
I am confused about u= 0 on x= 0 for t>0
0=t+F(-t²)
thus it follows that u(x,t)=t-√(t²+x)
"at" x=o
But what does it mean when is states this is only for t>0. For all t>0 when x is zero u will be zero anyway.
The answer shows that the funcrtion u is equal to what i have solved for t²>x,
and i can see that in this case the square root is defined(ie not √-)
Then from the second condition the solution shows that u=1+t for t²≤x
I would basically like to understand all this properly as i always get a bit confused with inequalities. How did they apply the second condition and inequality to get the second function and what was the reasoning?

 

[lippyka]

Also why did they state "for t>0" when it follows from the first condition that u=0 at x=0 for all t>0.