While studying differential equations, I have noticed that several solution techniques are nearly identical. Reduction of order, variation of parameters (first-order version), and transforming Riccati equations into Bernoulli equations, probably among others, all share the same basic structure:
1) Find a particular solution, either to the ODE or the associated homogeneous ODE, $y_p(x)$ or $y_h(x)$, respectively
2) Accordingly, plug some function $f(y_p(x), v(x))$ or $f(y_h(x), v(x))$ into the original ODE for $y$ and simplify, which generates a new ODE
3) Solve the new ODE
4) Unwind the dependent variable back to $y$
Is there a way to think more generally about applying this structure to solve ODEs, rather than as a collection of separate techniques? For example, in 1, the Riccati technique seeks a non-homogeneous particular solution $y_p$, while the reduction of order and first-order variation of parameters techniques seek a homogeneous particular solution $y_h$. In 2, the Riccati technique suggests $f = y_p + v$, the action of adding $v$ to the obtained solution, while the reduction of order and first-order variation of parameters techniques suggest $f = vy_h$, the action of multiplying the obtained solution by $v$. Is there a process for making these subtle choices manually, rather than memorizing/referencing the technique-specific prescriptions?
Also, is getting these choices "correct" essential to the accuracy of the final, general solution, or are they merely correct or incorrect by virtue of whether the new ODE in 3 is easier or harder to solve than the original ODE? It seems to be the case that choosing $f = y_h + v$ in a first-order variation of parameters problem yields a new ODE identical to the original ODE (with $y$ replaced by $v$), and is thus mathematically legal, just not helpful. On the other hand, when I tried to solve $y' - 4y = e^x$ by choosing $f = y_h\ ^v = e^{4vx}$, the action of raising the obtained solution $e^{4x}$ to the power of $v$, I obtained the new ODE $v' = \frac{e^{x - 4vx} - v + 4}{4x}$. After too many pages of algebra to post, I very much failed to verify the accuracy of this new ODE by solving $e^{4vx} =$ [the correct solution to the original ODE] $= Ce^{4x} - \frac{e^x}{3}$ for $v$ and plugging it into the new ODE. Was this the right approach to verify the accuracy of the new ODE, and should it have worked? If so, or more generally, if the new ODEs such as $v' = \frac{e^{x - 4vx} - v + 4}{4x}$ obtained by plugging a nonstandard $f$ into the original ODE are mathematically legal, is there a way to reverse the process, given the problem $v' = \frac{e^{x - 4vx} - v + 4}{4x}$ to find the original ODE $y' - 4y = e^x$, thus turning a difficult ODE into a simpler one?