If I have a differential equation on the form

$$y = y' \cdot c_1$$

can I freely solve for $y'$ and use the solution for

$$y' = y \cdot c_2$$

where $c_2 = \frac{1}{c_1}$?


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Yes, of course. Assuming that $c \in \mathbb R$ is a constant, then if $c \neq 0$ :

$$y = y' \cdot c \Leftrightarrow y' = y \cdot \frac{1}{c} \equiv y \cdot c$$

Since $c$ is an arbitrary constant, any expression of it will also be a constant, so you can always "manipulate" it to be just $c$. Note that only if you have some certain restrictions for $c$, then you will need to take these in mind on how they affect the expression $1/c$.


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