Question

dx/dt=k(a-x)

Answer 1

\[\large \frac{ dx }{ dt } = k(a-x)\] use separation of variables \[\large \frac{ dx }{ a-x } = k dt\] now integrate both sides \[\large \int\limits \frac{ dx }{ a-x } =\int\limits k dt\] \[\large -\ln ( a-x ) = kt + C\]

Answer 2

To finish, multiply it all by -1 first, then take the power of e on both sides. Note that x < a.