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$The general solution is implicit and found using the method of characteristics.$

$The general solution requires an integration and the introduction of one arbitrary constant for each, resulting in an implicit solution.$

$y = \int g(x) \, dx + C$, where $C$ is an arbitrary constant

$y = Ce^{-\int P(x) \, dx}$, where $C$ is an arbitrary constant

$y = e^{-\int P(x) \, dx}\left(\int Q(x)e^{\int P(x) \, dx}\, dx + C\right)$, where $C$ is an arbitrary constant

The form of the solution depends on the discriminant $b^2 - 4a$.

$y = C_1\cos(\sqrt{a}x) + C_2\sin(\sqrt{a}x)$, where $C_1$ and $C_2$ are arbitrary constants and $a>0$

$y = C_1e^{\sqrt{a}x} + C_2e^{-\sqrt{a}x}$, where $C_1$ and $C_2$ are arbitrary constants and $a>0$

The form of the solution depends on the discriminant $b^2 - 4a$.