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.
Best Answer
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 $$ y^2(x) = \left[f(x)\right]^2 = 2 \int a(x) dx + C $$