Chapter 4.2, Problem 26P

Consider the equation $y\text{'}\text{'}-y\text{'}-2y=0$

(a). Show that $y_1(t)=e^{-t}$ and $y_2(t)=e^{2t}$ form a fundamental set of solutions.

(b). Let $y_3(t)=-2e^{2t}$, $y_4(t)=y_1(t)+2y_2(t)$, and $y_5(t)=2y_1(t)-2y_3(t)$

Are $y_3(t),y_4(t)$ and $y_5(t)$ also solution of the given differential equation? $$

(c). Determine whether each of the following pair forms a fundamental set of solution:

$[y_1(t),y_3(t)]$; $[y_2(t),y_3(t)]$; $[y_1(t),y_4(t)]$; $[y_4(t),y_5(t)].$