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The set of solutions of $(E): y' + a(x)y = 0$ ($a\,:\mathbb{R}\,\rightarrow\,\mathbb{R}$ continuous function) is a one-dimensional vector space.

If $f(x) = e^{-\int_0^x a(t)\,\mathrm{d}t}$ is solution of $(E)$, why the general solution of $(E)$ is of the form $Cf(x)$ ($C\,\in\,\mathbb{R}$)?

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    Any nonzero element of a one-dimensional vector space spans the space...2012-09-16
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    @Hans Lundmark: so the general solution is of the form $Cf(x)$ because $f(x)$ is in the vector space, and $Cf(x)$ spans it because it is a one-dimensional space?2012-09-16
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    @DavidRobertJones: The general solution is of the form $Cf(x)$ because it can span the space. It is a non zero vector as you found it above.2012-09-16

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