a probleme about transformation of homogeneous SDE

I have a question about the characteristics of homogeneouse SDE:

$$dX_t=\beta(X_t)+\alpha(X_t)dW_t\ \ (1)$$

where $W$ implies a standard Brownian motion.

To be more specific, given a general SDE:

$$dY_t=\mu(t,Y_t)dt+\sigma(t,Y_t)dW_t\ \ (2)$$

Could we find a bijective transformation $u: X_t\rightarrow Y_t$ such that $X_t$ satisfies a homogenous SDE as $(1)$. I have a natural idea: set $u=u(t,x)$ then the Ito's formula gives

$$dY_t=(\frac{\partial u}{\partial t}+\beta(X_t)\frac{\partial u}{\partial x}+\frac{1}{2}\sigma^2(X_t)\frac{\partial^2 u}{\partial x^2})dt+\beta(X_t)\frac{\partial u}{\partial x}dW_t$$

Hence we have

$$\frac{\partial u}{\partial t}+\beta(x)\frac{\partial u}{\partial x}+\frac{1}{2}\sigma^2(x)\frac{\partial^2 u}{\partial x^2}=\mu(t,u(t,x))$$

$$\beta(y)\frac{\partial u}{\partial y}=\sigma(t,u(t,x))$$

In other words, under which conditions of $(\mu,\sigma)$ we could have a solution $(u,\beta,\alpha)$?

The motivation of this question comes from an algorithm exact simulation which gives a method to simulate exactly the distribution of $X_T$ of $(1)$, so I would like to know whether we can generalize this idea. Thanks a lot for your discussions!

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