In each of Problems $8$ through $11,$ verify that the given functions are solutions of the differential equation, and determine their Wronskian.

$8.y^{(4)}+a^2{y}^{\text{'}\text{'}}=0$;$1,t,cos(at),sin(at)$

$9.y^{\text{'}\text{'}\text{'}}-3y^{\text{'}\text{'}}-y^{\text{'}}+3y=0$;$e^t,e^{-t},e^{3t}$

$10.t{y}^{\text{'}\text{'}}-y^{\text{'}}=0$;$1,t^2$

$11.y^{(4)}+9y^{\text{'}\text{'}}=0$;$1,t,cos3t,sin3t$

$12.$ a$.$ Show that $W[5,co{s}^{2}t,cos(2t)]=0$ for all $t$ by directly evaluating the Wronskian.

b$.$ Establish the same result without direct evaluation of the Wronskian.