Define the linear stability domain for a numerical method to solve $y^{\prime}=f(t, y)$. What is meant by an $A$-stable method? Briefly explain the relevance of these concepts in the numerical solution of ordinary differential equations.

Consider

$$y_{n+1}=y_{n}+h\left[\theta f\left(t_{n}, y_{n}\right)+(1-\theta) f\left(t_{n+1}, y_{n+1}\right)\right]$$

where $\theta \in[0,1]$. What is the order of this method?

Find the linear stability domain of this method. For what values of $\theta$ is the method A-stable?