Characteristic equation:
[tex]64r^2+48r+8=0[/tex]
[tex]64\left(r^2+\dfrac34r+\dfrac18\right)=0[/tex]
[tex]\left(r+\dfrac12\right)\left(r+\dfrac14\right)=0\implies r=-\dfrac12,r=-\dfrac14[/tex]
This gives a general solution of
[tex]y_1=C_1e^{-1/2t}+C_2e^{-1/4t}[/tex]
With the given initial conditions, you get
[tex]y_1(0)=1\implies 1=C_1+C_2[/tex]
[tex]{y_1}'(0)=0\implies 0=-\dfrac12C_1-\dfrac14C_2[/tex]
[tex]\implies C_1=-1,C_2=2[/tex]
so that the first particular solution is
[tex]y_1(t)=-e^{-1/2t}+2e^{-1/4t}[/tex]
With the second set of initial conditions, you would have
[tex]y_2(0)=0\implies 0=C_1+C_2[/tex]
[tex]{y_2}'(0)=1\implies 1=-\dfrac12C_1-\dfrac14C_2[/tex]
[tex]\implies C_1=-4,C_2=4[/tex]
so that the second particular solution is
[tex]y_2(t)=-4e^{-1/2t}+4e^{-1/4t}[/tex]
The Wronskian of the two solution is
[tex]W(y_1,y_2)=\begin{vmatrix}y_1&y_2\\{y_1}'&{y_2}'\end{vmatrix}=\begin{vmatrix}-e^{-1/2t}+2e^{-1/4t}&-4e^{-1/2t}+4e^{-1/4t}\\-\frac12e^{-1/2t}-\frac12e^{-1/4t}&2e^{-1/2t}-e^{-1/4t}\end{vmatrix}[/tex]
[tex]W(y_1,y_2)=e^{-3/4t}[/tex]
