I'm trying to solve this Cauchy problem, which appears to be rather basic, but it seems like I'm missing something.
The problem is as follows:
$ u_x+u_t+u=0 \\ 0<\alpha*t<x<\infty \\ \alpha >0 , \ \alpha\neq1 \\ u(\alpha*t,t)=1, t>0\\u(x,0)=0,x>0 $
What I did was using the following sets of equations and initial conditions: $ \frac {dx}1=\frac {dt}1=\frac {du}{0-1*u}=ds \\ \Gamma_{s=0}:\{x=\tau, t=\tau/\alpha, u_0=1 \} $
After some integration and algebra, I reached the answer $ u(x,t)=e^{-\frac12(x+t+(t-x)(\frac{\alpha+1}{\alpha-1}))} $, but I can't seem to confirm whether this was the right way to do this, or whether the answer is correct, especially considering the fact I haven't used the initial condition $u(x,0)=0,x>0 $ anywhere.
Any help on how to approach this sort of problems? I've read various solutions of Cauchy problems, but couldn't find any that had two initial conditions rather than an initial curve of sorts.