Consider the dynamic programming problem:

$ma{x}_{y_i\text{i}\text{n}\text{R}}[{\displaystyle \underset{t=0}{\overset{T-1}{}}}-e^{-y_t}-e^{-x_T}],x_{t+1}=2x_t-y_t,t=0,1,$dots,$T-1$

where $,>0$ and $x_0$ is given exogenously.

$($a$)$ Compute $V_T(x),V_{T-1}(x)$ and $V_{T-2}(x)$ using the recursive formulation method.

$($b$)$ Solve the dynamic programming problem using the Euler equation method.