general relativity - Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options:

1. 
   

   using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $.

   
   



2. 
   

   using the fact that for a photon $ds^2=0$ to get: $0=-dt^2+e^tdx^2$ and then: $dt=\pm e^{t/2} dx$.

   
   

The problem is that (1) gives me $x=ae^{-t}+b$ and (2) gives me $x=ae^{-t/2} +b$.