Can you place any variable in a function, and will it remain the same function?

I came across this question about homogenous differential equations, which has me scratching my head because it seems like its doing magic. The problem is as follows:

A particular solution of the differential equation $xfrac{dv}{dx}=f(v)$ has $v=2$ when $x=e$. If $k$ is the value of $v$ when $x=1$, show that $int_k^2frac{1}{f(y)}dy=1$.

I started by using separation of variables to get that $$intfrac{1}{f(v)}dv=intfrac{1}{x}dx$$
$$intfrac{1}{f(v)}dv=ln(x)$$
$$int_k^2frac{1}{f(v)}dv=left[ln(x)right]_1^e=1$$

Here is when I no longer understand what am I supposed to do. Should I just say that it does not really matter which variable is being used in the function $f(x)$ because it will always be the same? I also don’t really know if the question assumes that $y=vx$, as is used in v-substitution, but it was never explicitly stated.