15.1.35 Theory of random processes

Problem: Find the expected value \( m_{Y}(t) \) and the spectral density \( S_{Y}(\omega) \) of the solutions of the equation \( Y^{\prime \prime}(t)+2 Y^{\prime}(t)+2 Y(t)=X^{\prime}(t)+X(t), \quad \) where \( X(t) \) is the stationary function, and \( m_{X}(t)=1 \), \( K_{X}(\tau)=e^{-|\tau|}(1+|\tau|) \).