Radiocarbon dating differential equation
Thus, we have will have an additional force, Here we let \(m = 1\text\) \(b = 3\text\) and \(k = 2\text\) We will learn how to solve equations of the form \(mx'' bx' kx = 0\) in Chapter 4, but let us assume that the solution is of the form \(x(t) = e^\) for now.
In this case, Of course, if we have a very strong spring and only add a small amount of damping to our spring-mass system, the mass would continue to oscillate, but the oscillations would become progressively smaller.
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, Japan has experienced negative growth in recent years.
The equation \(d P/dt = k P\) can also be used to model phenomena such as radioactive decay and compound interest—topics which we will explore later.
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.