Weakly stationary AR process

Question

I have the following AR(2) process: $$Y_t=Y_{t-1}-0.5Y_{t-2}+\epsilon_t,$$ where $\epsilon_t$ is a white noise with $0$ expectation and variance equal to $1$. Is this procecss weakly stationary?

In order to check if it is weakly stationary I should check that

1. $\mathbb E(Y_t)$ is independent from $t$

2. $var(Y_t)$ is independent from $t$

3. $cov (Y_t, Y_{t+k})=0$ for every $t, k: k\neq 0$

Using the first two conditions I just proved that if the process is weakly stationary then $\mathbb E(Y_t)=0$ and $var (Y_t)=\frac{1}{0.025}$ but I don't know how to check if the process is weakly stationary. Could someone help me?