Location-independent differential equation/Constant distance between solutions/Understanding/Exercise

Convince yourself that in a location-independent differential equation (i.e. ${}f(t,y)$ does not depend on ${}y$ ) the difference between two solutions ${}y_{1}$ and ${}y_{2}$ does not depend on time, that is ${}y_{1}(t)-y_{2}(t)$ is constant.

Show with an example that this may not happen in a time-independent differential equation.