Inspired by this answer, if we have the differential equation
$${y_1}'= f_{11}(x)y_{1}+f_{12}(x)$$
Under which conditions can we be sure that the iteration by factoring $f_{11}\cdot\left(y_1+ \frac{f_{12}}{f_{11}}\right)$ followed by substitution $y_2 = y_1+\frac{f_{12}}{f_{11}}$, rewriting a new differential equation in terms of $y_2$: $${y_2}'= f_{21}(x)y_{2}+f_{22}(x)$$ and iterating will finish at some $n$ at a form:
$${y_n}' = f_{n1}(x)y_n $$ ( Which can then be solved with the chain rule and we can backtrack a solution $n\to n-1 \to \dots\to 1$ )