Assume two systems for which the following differential equations hold between their input and output signals.
$$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$
$$\dfrac{dy(t)}{dt}=v(t)u(t)$$
Also, assume that the input process $x(t)$ is white Gaussian noise (with mean zero) such that its correlation function is $R_x(\tau)=2b \delta(\tau)$ where $a,b$ are constants.
a) Calculate the value of the correlation function of the output process at zero, $R_y(0)$
b) Is $y(t)$ a WSS process?
So, this is how I imagine this question could be solved: 1. calculate $R_y$. Then measure it on $0$. (How? I do not know). To prove that a process is WSS (wide-sense stationary), I should prove that its mean (in this case $\mathbb{E}\{y(t)\}$ is a fix number like $\eta$ and prove that its autocorrelation only depends on $\tau=t_1-t_2$ (in other words, $\mathbb{E}\{y(t+\tau)y^*(t)\}=R(\tau)$). Since $\tau$ is actually the distance between $t$ and $t+\tau$, we would write $R(\tau)$ like this: $$R(\tau)=\mathbb{E}\{y(t+\frac{\tau}{2})y^*(t-\frac{\tau}{2})\}$$
However, I am new to all of these concepts and do not know how to do all of the things I mentioned. An answer with a little bit of detail would be very nice of you. Thanks in advance.