William A. Adkins - Ordinary Differential Equations

Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA

Example 1 (Newtons Law of Heating and Cooling).

Suppose we are interested in the temperature of an object (e.g., a cup of hot coffee) that sits in an environment (e.g., a room) or space (called, ambient space) that is maintained at a constant temperature T a. Newtons law of heating and cooling states that the rate at which the temperature T ( t )of the object changes is proportional to the temperature difference between the object and ambient space. Since rate of change of T ( t ) is expressed mathematically as the derivative, T ( t ), Newtons law of heating and cooling is formulated as the mathematical expression where r is the constant of proportionality. Notice that this is an equation that relates the first derivative T ( t ) and the function T ( t ) itself. It is an example of a differential equation. We will study this example in detail in Sect..

Example 2 (Radioactive decay).

Radioactivity results from the instability of the nucleus of certain atoms from which various particles are emitted. The atoms then decay into other isotopes or even other atoms. The law of radioactive decay states that the rate at which the radioactive atoms disintegrate is proportional to the total number of radioactive atoms present . If N ( t ) represents the number of radioactive atoms at time t , then the rate of change of N ( t ) is expressed as the derivative N ( t ). Thus, the law of radioactive decay is expressed as the equation As in the previous example, this is an equation that relates the first derivative N ( t ) and the function N ( t ) itself, and hence is a differential equation. We will consider it further in Sect..

Example 3 (Newtons Laws of Motion).

Suppose s ( t ) is a position function of some body with mass m as measured from some fixed origin. We assume that as time passes, forces are applied to the body so that it moves along some line. Its velocity is given by the first derivative, s ( t ), and its acceleration is given by the second derivative, s ( t ). Newtons second law of motion states that the net force acting on the body is the product of its mass and acceleration . Thus, Now in many circumstances, the net force acting on the body depends on time, the objects position, and its velocity. Thus, F { net}( t ) = F ( t , s ( t ), s ( t )), and this leads to the equation A precise formula for F depends on the circumstances of the given problem. For example, the motion of a body in a spring-body-dashpot system is given by , where and k are constants related to the spring and dashpot and f ( t ) is some applied external (possibly) time-dependent force. We will study this example in Sect.. For now though, we just note that this equation relates the second derivative to the function, its derivative, and time. It too is an example of a differential equation.

Each of these examples illustrates two important points: