# Radiocarbon dating differential equation

It is not too difficult to see that \(P(t) = Ce^\) is a In addition, if we know the value of \(P(t)\text\) say when \(t = 0\text\) we can also determine the value of \(C\text\) For example, if the population at the time \(t = 0\) is \(P(0) = P_0\text\) then Of course, it is important to realize that this is only a model.If \(t\) is small, our model might be reasonably accurate.In other words, our harmonic oscillator would be ¶Some situations require more than one differential equation to model a particular phenomenon.We might use a system of differential equations to model two interacting species, say where one species preys on the other.).We can use the Suppose we have a pond that will support 1000 fish, and the initial population is 100 fish.In order to determine the number of fish in the lake at any time \(t\text\) we must find a solution to the initial value problem ¶Sometimes it is necessary to consider the second derivative when modeling a phenomenon.

For example, we might add a dashpot, a mechanical device that resists motion, to our system.If we also assume that the population has a constant death rate, the change in the population \(\Delta P\) during a small time interval \(\Delta t\) will be is one of the simplest differential equations that we will consider.The equation tells us that the population grows in proportion to its current size.Suppose that we have a mass lying on a flat, frictionless surface and that this mass is attached to one end of a spring with the other end of the spring attached to a wall.We will denote displacement of the spring by \(x\text\) If \(x \gt 0\text\) then the spring is stretched. If \(x = 0\text\) then the spring is in a state of equilibrium (Figure 1.1.4).