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I'm a bit confused about why these types of problems are presented the way they are.

First order linear differential equation:

$$dy/dt + p(t) y = g(t);$$

First order separable differential equation:

$$M(t) + N(y)dy/dt = 0.$$

How are these related? I think if we take $N(y) = 1/y, M(t) = p(t)$ and $g(t)=0$, then the first order linear differential equation is separable. Is there some sort of deeper connection here that I'm missing?

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$p(t)$, $g(t)$, $M(t)$, $N(y)$ are predefined functions. Usually they depend on the actual problem you are trying to solve. In the very specific case you described you are indeed right.

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