A Solving dS/dt = −kS

In Section 4.3, we showed that the survival probability for a memoryless process satisfies \(\mathrm{d}S/\mathrm{d}t = -kS(t)\). Here we solve this differential equation formally. Separating variables and integrating with the initial condition \(S(0) = 1\):

\[ \int_1^{S(t)} \frac{\mathrm{d}S'}{S'} = \int_0^t -k\,\mathrm{d}t' \]

\[ \ln S(t) = -kt \]

\[ S(t) = \exp(-kt) \]