Finding a differential equation when a half life is known

Question

Does anyone know how I would write a differential equation for the following? I am not interested in the answer as such, I'm more interested in the steps and how to obtain the answer. I don't know how to write differential equations and I really need to understand how to.

Please note that I am not demanding an answer, I simply want to understand the steps in writing difference equations in general.

Here is an example of one I am trying to understand how to write: $C^{14}$ is a naturally occurring radioactive substance that is used in dating artifacts and fossil remains. It has a half life of 5730 years. Assuming the initial amount of $C^{14}$ is $C_0$, write a differential equation for the amount of $C^{14}$ after $n$ years, $C_n$.

Answer 1

This was posted on the assumption that a difference equation was required

You have $C_{n+5730}=\cfrac 12 C_n$ which can be written $$2C_{n+5730}-C_n=0$$ or$$2C_{n}-C_{n-5730}=0$$

Answer 2

The ODE for this problem is given by $$c(0) = c_0, c'(t) = \lambda c(t)$$ where $\lambda < 0$ is given by $\lambda = -\frac{\log(2)}{T_\frac{1}{2}}$ because $$c(t) = c_0 \left(\frac{1}{2} \right)^\frac{t}{T_{1/2}} = c_0 \exp( \frac{t}{T_{1/2}} \log(\frac{1}{2})) = c_0 \exp\left(-\frac{\log(2)}{T_{1/2}} t\right)$$

Answer 3

Hint: Every year some constant amount of $C^{14}$ decays: $C_n=\alpha C_{n-1}$. Find a general formula for $C_n$ and then find the value of $\alpha$. "It has a half life of 5730 years" means $C_{5730} = C_0/2$.

Answer 4

In order to write down an ODE, you need to know more about the problem. Indeed, an ODE for $C(t) =$[amount of C$^{14}$ at time $t$] is a very strong statement for $C$, since it constrains $C$ over the entire continuum $R_+ = \{t\ge0\}$. In particular, it is much stronger than the statement $C(5730)=\frac12 C(0)$, which constrains $C$ only on the discrete set $\{5730 n \,\vert\, n\in\mathbf{N} \}$, so this last statement is insufficient to derive an ODE for $C$.

Now, implicit in the term 'radioactive substance' is the additional modeling assumption that the amount of C$^{14}$ decays at a rate proportional to itself. (This is necessarily an independently verifiable experimental fact - or abstraction thereof - which informs our modeling. Without such knowledge, molding at the continuous time-level is plainly impossible, as answers in the current thread not assuming this fact also demonstrate.) A mathematical formulation of this statement is $$ \mbox{[rate of decay]} = -\frac{dC(t)}{dt} = \lambda C(t) $$ or, if you prefer, $$ 0 < C(t) - C(t+\delta t) \approx \lambda C(t) \delta t , \quad\mbox{with the approx. equality becoming exact in the limit $\delta t\downarrow0$.} $$ (Dividing by $\delta t$ & sending $\delta t\downarrow0$ yields the above ODE, of course.)

Supplementing this w/ the IC $C(0)=C_0$ yields an IVP (initial-value problem) w/ a unique solution.

(Your feedback would be appreciated; I'm trying my hand on explaining modeling here, & I will invariably use this explanation on some student body sooner or later.)