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).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. The subject of differential equations is one of the most interesting and useful areas of mathematics.

The company noticed that the number of pelts varied from year to year and that the number of lynx pelts reached a peak about every ten years .

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.

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.   The number of trout will be limited by the available resources such as food supply as well as by spawning habitat.

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