So this is an answer but reading and reading. I still have no idea how to determine the Wronskian y1 and y2. And when do I use it? I mean in general how do i find y1 and y2? I know the answer for this but I am asking in general if another question requires me to use Wronskian.
On
If $y_1$ and $y_2$ are the solution to a homogeneous differential equation the Wronskian is given by:
$$ \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \\ \end{vmatrix} $$
In case of $y''+y= 0$ the characteristic equation is $r^2+1=0$ which implies roots are $r=\pm i$ which means the solution is $c_1 \cos t+ c_2 \sin t$. So $y_1 =\cos t$ and $y_2= \sin t$ and so the Wronskian is
$$ \begin{vmatrix} \cos t & \sin t \\ -\sin t & \cos t \\ \end{vmatrix} $$
Which evaluates $\cos^2t+\sin^2t=1$ which means the Wronskian is non zero and hence the solution are independent.
$W$ is the Wronskain it's a determinant. $$W (y_1,y_2)=y'_1y_2-y_1y'_2$$ $$W(\sin t, \cos t)=\cos t \cos t -\sin t (-\sin t ) \\ W=\cos^2 t+\sin^2 t=1 \ne 0$$ $y_1,y_2$ here are solutions to the homogeneous differential equation: $$y''+y=0$$ $$r^2+1=0 \implies r=\pm i$$ $$y=c_1y_1+c_2y_2=c_1 \cos t+c_2\sin t$$