910. A stationary random function $X(t)$ with mean $m_{x}=5$ is input to a linear stationary dynamic system described by the equation $Y^{\prime}(t)+2 Y(t)=5 X^{\prime}(t)+$ $+6 X(t)$. Find the mean of the random function $Y(t)$ at the output of the system in the steady state (after the transient process has decayed).
Solution. Let's equate the mathematical expectations of the left and right parts of the given differential equation:

$$
\begin{aligned}
M\left[Y^{\prime}(t)+2 Y(t)\right]= & M\left[5 X^{\prime}(t)+6 X(t)\right], \text{ or } M\left[Y^{\prime}(t)\right]+2 m_{y}= \\
& =5 M\left[X^{\prime}(t)\right]+6 m_{x} .
\end{aligned}
$$

By the condition, $X(t)$ and $Y(t)$ are stationary functions, and the mathematical expectation of the derivative of a stationary function is zero, so
$2 m_{y}=6 m_{x}$. Therefore, the required mathematical expectation $m_{y}=3 m_{x}=$ $=3 \cdot 5=15$.