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).