it is hard to get a close form formula of the SDE， please search the key word CKLS model in 《Intrest rate model theory and method》，there are useful information.

搞错了。。。

Here you go:

the general solution to 1-dimensional linear SDE, not sure it will work out your question or not. But this SDE is nested in the class of 1-dimensional linear SDE.



someone would call this Brennan-Schwartz model, I prefer the name of lognormal vasicek :-)

best,