I am a little rusty with some calculus and need some help with the follow equation:
\begin{equation} \int\dfrac{f'(x)}{f(x)}dx = \int\dfrac{1}{-x+x^2y}dx \end{equation}
Where $y$ is a constant. My idea is to use some kind of $U$ substitution as I know $\int\frac{dv}{v} = \ln(v)$. This gives:
\begin{align} \ln(f(x)) =& \int\dfrac{1}{-x+x^2y}dx \\ \end{align}
Then I see to solve for $f(x)$ I can exponentiate. To solve the right integrand I would first do partial fraction decomposition which gives: $A=-1$ and $B=y$. Then I have:
\begin{align} \ln(f(x)) = \int\dfrac{-1}{x}dx + \int\dfrac{y}{-1+xy}dx \\ \end{align}
Then I get:
\begin{equation} f(x) = x+-1+xy = -1+(y+1)x \end{equation}
Is this correct?