First, we divide both sides of the equation by $t^2$:

$$y''+{t+2\over t}y'+{t+2\over t^2}y=0$$

Now the given second-order differential equation has the form:

$$L[y]=y''+p(t)y'+q(t)y=0$$

Noting if we let $p(t)={t+2\over t}$ and $q(t)={t+2\over t^2}$, then $p(t)$ is continuous everywhere except at $t=0$, and $q(t)$ is also continuous everywhere except at $t=0$. $\\$

Therefore, by Abel's Theorem: the Wronskian $W[y_1,y_2](t)$ is given by

$$\begin{align}W[y_1,y_2](t)&= cexp(-\int{p(t)dt})\\&=cexp(-\int{{t+2\over t}dt})\\&=ce^{t+2ln|t|}\\&=ct^2e^t\end{align} $$