Question #41805

The probability $P(t)$ that a quantum system remains in its initial state at time $t \ge 0$ satisfies the differential equation $\frac{dP}{dt} = -\frac{P^2}{2tP + 1}$ with the initial condition $P(0) = 1$. A researcher measures the state at $t=2$ and correctly determines the probability $P(2)$. To predict the probability at $t=12$, the researcher incorrectly assumes the system's decay is memoryless, satisfying $P(t_1 + t_2) = P(t_1)P(t_2)$ for all $t_1, t_2 \ge 0$. Let $P_{est}$ be the researcher's estimated probability at $t=12$, and $P_{true}$ be the actual probability at $t=12$ governed by the differential equation. Find the value of $\frac{P_{true}}{P_{est}}$.