the integral curve is： dt = dx/x

if I integrate both sides, I get t - lnx = c, so the general solution is u = f(t-lnx)

however, it I move x to the left side: xdt = dx, and integrate both sides, I get xt - x = c. You cannot do this, since in ODE , describing integral curves, $x$ and $t$ are not independent. V.I.

the general solution becomes u = f(xt-x)

I feel the second approach is wrong, but I cannot tell where did I do wrong.