Consider the first order system

$$\frac{d \mathbf{x}}{d t}-A \mathbf{x}=e^{\lambda t} \mathbf{v}$$

to be solved for $\mathbf{x}(t)=\left(x_{1}(t), x_{2}(t), \ldots, x_{n}(t)\right) \in \mathbb{R}^{n}$, where $A$ is an $n \times n$ matrix, $\lambda \in \mathbb{R}$ and $\mathbf{v} \in \mathbb{R}^{n}$. Show that if $\lambda$ is not an eigenvalue of $A$ there is a solution of the form $\mathbf{x}(t)=e^{\lambda t} \mathbf{u}$. For $n=2$, given

$$A=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad \lambda=1, \quad \text { and } \quad \mathbf{v}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right)$$

find this solution.