(a) By considering eigenvectors, find the general solution of the equations

$$\begin{aligned} &\frac{d x}{d t}=2 x+5 y \\ &\frac{d y}{d t}=-x-2 y \end{aligned}$$

and show that it can be written in the form

$$\left(\begin{array}{l} x \\ y \end{array}\right)=\alpha\left(\begin{array}{c} 5 \cos t \\ -2 \cos t-\sin t \end{array}\right)+\beta\left(\begin{array}{c} 5 \sin t \\ \cos t-2 \sin t \end{array}\right)$$

where $\alpha$ and $\beta$ are constants.

(b) For any square matrix $M$, $\exp (M)$ is defined by

$$\exp (M)=\sum_{n=0}^{\infty} \frac{M^{n}}{n !}$$

Show that if $M$ has constant elements, the vector equation $\frac{d \mathbf{x}}{d t}=M \mathbf{x}$ has a solution $\mathbf{x}=\exp (M t) \mathbf{x}_{0}$, where $\mathbf{x}_{0}$ is a constant vector. Hence solve $(\dagger)$ and show that your solution is consistent with the result of part (a).