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Say I have a second order linear homogeneous differential equation with variable coefficients:

$\displaystyle \frac{d^2 y}{dx^2}+ xy=0$

Why am I unable to use the standard technique of subbing in $y=e^{mx}$ that is used when we have constant coefficients?

Is it because $m$ would become a variable and $\frac{dy}{dx}$ is no longer $me^{mx}$?

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Or rather because if $m$ is constant, $e^{mx}$ is not a solution: plug it in and you'll see why.

The fundamental solutions of this differential equation happen to be non-elementary functions (Airy functions).

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