I find out the solution of Ornstein-Uhlenbeck equation is
My problem is how to find and
It seems like it is easy to find the mean and variance.But I was trying and got no solution
All you need to know is $\displaystyle E[\int_0^t f(s) dB_s]=0$, because this integral is a martingale (as a function of $\displaystyle t$), and $\displaystyle E[\left(\int_0^t f(s) dB_s\right)^2]=\int_0^t f(s)^2 ds$, as follows from the definition of the stochastic integral.
It depends on what is $\displaystyle X_0$. If it is constant, then you are correct (and $\displaystyle E[X_t]=X_0e^{\mu t}$). If it is a random variable, then it must appear in the variance: $\displaystyle X_t-E[X_t]=(X_0-E[X_0])e^{\mu t} +\int_0^t (\cdots)$ and both terms are independent and centered hence $\displaystyle E[(X_t-E[X_t])^2]={\rm Var}(X_0)e^{2\mu t}+E[(\int_0^t(\cdots))^2]$ =$\displaystyle {\rm Var}(X_0)e^{2\mu t}+\frac{\sigma^2}{2\mu}(e^{2\mu t}-1)$. I think.