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.