Chapter 9.6, Problem 16E

Show that $x_1\left(t\right)$ and $x_2\left(t\right)$ given by equations (4) and (5) can be obtained as linear combinations of $w_1\left(t\right)$ and $w_2\left(t\right)$ given by equations (2) and (3). [Hint: Show that

$x_1\left(t\right)=\frac{w_1\left(t\right)+w_2\left(t\right)}{2}$, $x_2\left(t\right)=\frac{w_1\left(t\right)-w_2\left(t\right)}{2i}$].

$w_1\left(t\right)=e^{r_{1}t}z=e^{\left(\alpha +i\beta \right)t}\left(a+ib\right)$

$w_2\left(t\right)=e^{r_{2}t}\bar{z}=e^{\left(\alpha -i\beta \right)t}\left(a-ib\right)$

$x_1\left(t\right):=e^{\alpha t}\cos \beta ta-e^{\alpha t}\sin \beta tb$

$x_2\left(t\right):=e^{\alpha t}\sin \beta ta+e^{\alpha t}\cos \beta tb$