I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble understanding why we choose certain substitutions.
For example
1. Solve the following differential equation:
$dX_t = (\beta - \alpha X_t)dt + \sigma dB_t, \quad X_0 = x_0$
Question 1: Here we set $Y_t = e^{\alpha t}X_t$, find $dY_t$ and then solving the rest of the SDE is easy. Why do we set $Y_t = e^{\alpha t}X_t$? and how do we know how to choose it?
2. and the following: $dX_t = [1-\ln(X_t)]X_tdt + \sigma X_t dB_t$
Here we first set $Y_t = \ln(X_t)$ find
$dY_t = (1-\frac{\sigma^2}{2}-Y_t)dt + \sigma dB_t$
then we need to set $Z_t = e^t Y_t$ and solve:
$dZ_t = e^t(1 - \frac{\sigma^2}{2})dt + e^t\sigma dB_t$
Integrate $dZ_t$ to find $Z_t$ and lastly put it back to $X_t$ giving
$X_t = e^{Y_t} = e^{e^{-t}Z_t}$
Question 2: Again how do we know which substitutions to choose?
Question 3: Is there any general rule for how to set the substitutions?
Thanks!