I came across a differential equation:
$\frac{dy}{dx}=\frac{\sin(\log x)}{\log y}$. Here is what I tried to do:
I transformed it into this form $\log y dy=\sin(\log x)dx$ i.e. $\int \log y dy=\int \sin(\log x)dx\dots(2)$ and after that I used integration by parts to finish off the problem.
However,I was told by my teacher that it should instead be $\int \log y dy=\int \sin(\log x)dx+C$ where $C$ is a constant of a integration.I argued that the integration had not yet been carried out and so there was no need for the constant.(and I was told it $had$ to be there.)
Can anyone please convince me why my teacher is right and I wrong?
Thanks.