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Consider the linear differential equation $ \frac{dy}{dx}+p(x)y=q(x).$ 1. What conditions may be imposed on functions $p(x)$ and/or $q(x)$ so that it could directly be solved by variable separable method rather than integrating factor $e^{\int p(x) dx}$. 2. In the case if it can be solved by both variable separable and integrating factor method, which would be more suitable?

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Comparing the general linear ODE and the separable linear ODE, \begin{equation} \frac{dy}{dx} + p(x) y = q(x) \quad \textrm{and} \quad \frac{dy}{dx} = f(x) (ay+b), \end{equation} we note that the linear ODE is separable if and only if \begin{equation} q(x) - p(x)y = f(x) (ay+b), \end{equation} with a new function $f$. This is the case if $q(x) = c p(x)$ for some constant $c \in \mathbb{R}$, since then we obtain $q(x) - p(x) y = p(x)(c-y)$.

The solution of a separable linear ODE \begin{equation} \frac{dy}{dx} + p(x) y = c p(x) \quad \Leftrightarrow \quad \frac{dy}{dx} = p(x)(c-y) \end{equation} via separation of variables or via an integrating factor is a matter of preference, and with both methods we obtain the general solution \begin{equation} y(x) = c + C e^{-P(x)}, \quad C \in \mathbb{R}, \end{equation} where $P$ is an antiderivative of $p$.

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