I have the following ODE:
\begin{equation} f'(x) = \frac{a}{x}f(x) \end{equation}
where $a$ is a constant, $x \in [0,1]$.
With the following initial condition:
\begin{equation} f'(1) = 0 \end{equation}
If I integrate both sides:
\begin{equation} f(x) = c\cdot x^{a} \end{equation}
However, this initial condition is implying that $a = 0$ and $f(x) = c$. Which means that $f'(x) = 0$ $\forall$ $\in$ $[0,1]$. But that cannot be true in my model.
Is there any other solution to this problem? Or it must be this one?