Consider the first order system

$$\frac{d \boldsymbol{v}}{d t}-B \boldsymbol{v}=e^{\lambda t} \boldsymbol{x}$$

to be solved for $\boldsymbol{v}(t)=\left(v_{1}(t), v_{2}(t), \ldots, v_{n}(t)\right) \in \mathbb{R}^{n}$, where the $n \times n$ matrix $B, \lambda \in \mathbb{R}$ and $\boldsymbol{x} \in \mathbb{R}^{n}$ are all independent of time. Show that if $\lambda$ is not an eigenvalue of $B$ then there is a solution of the form $\boldsymbol{v}(t)=e^{\lambda t} \boldsymbol{u}$, with $\boldsymbol{u}$ constant.

For $n=2$, given

$$B=\left(\begin{array}{ll} 0 & 3 \\ 1 & 0 \end{array}\right) \quad \lambda=2 \quad \text { and } \boldsymbol{x}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

find the general solution to (1).