 Solution

For $x_0>0$, separation of variables gives $1/x=kt+1/x_0$, hence $x(t)=x_0/(1+k{x}_{0}t)$. Setting $x(t)=x_0/2$ gives the half-lifeThis is the same reciprocal decay law as a second-order reaction with the specified disappearance rate coefficient. A positive initial value is needed for a positive half-time; $x_0=0$ instead gives the identically zero solution.