1. For the system shown below, where the autocorrelation function is $\phi_{x_2x_2}(\tau) = \sigma^2 e^{-\beta|\tau|}$,
show that $P(t) = \frac{\sigma^2}{\beta^2} \left[ \frac{2(\beta t - 1 + e^{-\beta t})}{\beta(1 - e^{-\beta t})} + \frac{\beta(1 - e^{-\beta t})}{\beta^2} \right]$ for the state vector taken as $x = [x_1 \quad x_2]^T$.
