Here's a fairly standard derivation of the exponential growth equation.
$\frac{\text{d}x}{\text{d}t} = kx$
$\int\frac{1}{x}\text{d}x = \int k\text{d}t$
$ln(x)=kt+C$
$x=C'\text{e}^{kt}$
Right?
But what if I don't separate my variables in step 2? I don't have to.
$\int \text{d}x = \int kx \text{d}t$
$x = kxt + C$
$(1-kt)x = C$
$x = \frac{C}{1-kt}$
Nothing like the same result! They can't both be correct, but each step looks sound to me. What's the error?