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In addition, the theory of the subject has broad and important implications.¶We begin our study of ordinary differential equations by modeling some real world phenomena. The subject of differential equations is one of the most interesting and useful areas of mathematics.We can describe many interesting natural phenomena that involve change using differential equations.To summarize, we say that the expression \(x(t) = Ce^\) is a ¶Not all populations grow exponentially; otherwise, a bacteria culture in a petri dish would grow unbounded and soon be much larger than the size of the laboratory.
If \(t\) is small, our model might be reasonably accurate.
Thus, we can consider the restorative force on the spring to be proportional to displacement of the spring from its equilibrium length, Suppose that we have a spring-mass system where \(m =1\) and \(k = 1\text\) If the initial velocity of the spring is one unit per second and the initial position is at the equilibrium point, then we have the following initial value problem, Now let us add a damping force to our system.
For example, we might add a dashpot, a mechanical device that resists motion, to our system.
The number of trout will be limited by the available resources such as food supply as well as by spawning habitat.
A small population of fish might grow exponentially if the pond is large and food is abundant, but the growth rate will decline as the population increases and the availability of resources declines.
For example, Japan has experienced negative growth in recent years.