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I'm sure this must be a duplicate somewhere on here and for that I apologize but I can't seem to find it.

But can anyone tell me or show me the general method of solving a differential equation of the form:

$$f'(x) = \frac{a(x)}{f(x)}.$$

So, variable coefficients. I'm just really confused that now I have to deal with a fraction! Any help would be great.

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Possibly this is separable? Let $y = f(x)$. Your relation implies $$ \frac{dy}{dx} = \frac{a(x)}{y} $$ which in turn means $$ \int y dy = \int a(x) dx $$ and the LHS yields $y^2/2$, so you get $$ y^2(x) = \left[f(x)\right]^2 = 2 \int a(x) dx + C $$

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Note that, by implicit differentiation, $\frac{d}{dx}[f(x)^2] = 2f(x)f^\prime(x)$, so your equation rearranges to $\frac{d}{dx}[f(x)^2]=2a(x)$. Integrating both sides gives $f(x)^2=C+2\int a(x) dx$, so $f(x) = \sqrt{C+2\int a(x) dx \ }$

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