a)
The coefficient of $y''$ is -1
then Wronskain is $Ce^{t}$
b)
Use homogeneous equation to find fundamental solutions
$y'''-y''+4y'-4y=0$
Then $r^3-r^2+4r-4=0$
Then $(r^2+4)(r-1)=0$
Then$ r_1=-2i, r_2=2i, r_3=1$
Then the solution is $y=C_1\cos2t+C_2\sin2t+C_3e^t$
So
$W(y_1, y_2, y_3)(t) = \begin{bmatrix} \cos2t&\sin2t&e^{t}\\-2\sin2t&2\cos2t&e^{t}\\-4\cos2t&-4\sin2t&e^{t}\\ \end{bmatrix}=10e^{t}$
This is consistent with what we get in part (a)
c) Use undetermined coefficients method
Assume $y(t) = At\cos{2t}+Bt\sin{2t}$
$y'(t) = A\cos{2t}-2At\sin2t+B\sin2t+2Bt\cos2t$
$y''(t) = -4A\sin{2t}-4At\cos2t+8At\sin2t-8B\sin2t$
$y'''(t) = -12A\cos{2t}+8At\sin2t-12B\sin2t-8Bt\cos2t$
Plug into the equation,
We get $-8A-4B=8, 4A-8B=0$
we get $A=-\frac{4}{5}$
$B=-\frac{2}{5}$
Thus, $Y=C_1\cos2t+C_2\sin2t+C_3e^t-\frac{4}{5}t\cos2t-\frac{2}{5}t\sin2t$
