基于West&Hubbard微分方程教材碳定年案例，作者如何构建微分方程？

Alright, let's walk through exactly how West & Hubbard build the differential equation for carbon-14 dating in this example, tying directly to the context you provided:

• Step 1: Define the key variable
  The authors start by defining a clear, meaningful function: let $N(t)$ represent the total amount of $C^{14}$ remaining in the organism at time $t$. They set $t=0$ as the exact moment the organism dies—this is critical because death stops the organism from exchanging $C^{14}$ with its environment, breaking the equilibrium that kept $C^{14}$ levels stable during life.

• Step 2: Derive the rate of change using radioactive decay rules
  They lean on the fundamental physical law of radioactive decay: the rate at which a radioactive substance decays is directly proportional to the amount of the substance present.

  From the problem context: While the organism is alive, $C^{14}$ lost to decay is replaced by environmental exchange, so the net change in $N(t)$ is zero ($\frac{dN}{dt}=0$). After death, there's no more replacement—only decay, so $N(t)$ decreases over time.

  Translating this to math: Let $\lambda$ be the positive decay constant for $C^{14}$ (a value that quantifies how fast the isotope decays). The instantaneous rate of change of $N(t)$ will be a negative multiple of $N(t)$ (negative because the amount is decreasing).

• Step 3: Formulate the final differential equation
  Putting it all together, the authors write the differential equation that governs $C^{14}$ levels after the organism's death:
  $$\frac{dN}{dt} = -\lambda N(t)$$
  They also include the initial condition $N(0) = N_0$, where $N_0$ is the amount of $C^{14}$ in the organism at death—this is equal to the equilibrium concentration present during its life.

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