What is the time constant L/R of this circuit?
Let us re-check your solution through laplace method.

At t>=0,

$$\frac{2}{s} = L(sI(s) - I(0)) + I(s)R_{eq} $$
$$\implies \frac{2}{s} = sI(s)-1+I(s)$$
$$\implies \frac{2}{s}+1 = I(s)(s+1)$$
$$\implies \frac{2+s}{s(s+1)} = I(s)$$
$$\implies I(s) = \frac{2}{s}+\frac{-1}{s+1}$$
Taking inverse laplace,
$$I(t) = 2 - e^{-t} = 2(1-\frac {e^{-t}}{2})$$

Your solution is therefore correct.