1. Problems that lead to differential equations

A differential equation is a relation between a function and its derivatives. Differential equations form the language in which the basic laws of physical science are expressed. The science tells us how a physical system changes from one instant to the next. The theory of differential equations then provides us with the tools and techniques to take this short term information and obtain the long-term overall behavior of the system.

A dynamic physical system.

This pendulum illustrates how a physical system changes with time.

There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us.

(1)

 

To see that this is in fact a differential equation we need to rewrite it a little. First, remember that we can rewrite the acceleration, a, in one of two ways.

(2)

 

Where v is the velocity of the object and u is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position. So, with all these things in mind Newton’s Second Law can now be written as a differential equation in terms of either the velocity, v, or the position, u, of the object as follows.

(3)

(4)

 

So, here is our first differential equation.  We will see both forms of this in later chapters.

A differential equation (DE) is an equation involving derivatives of unknown function of one or more variables. If the unknown function depends on only one variable, the equation is called an ordinary differential equation (ODE). If the unknown function depends on more than one variable, the equation is called a partial differential equation (PDE).

Example 1: or is an ordinary differential equation since the function depends on only one variable .In the function , is called the independent variable, and is the dependent variable.

Example 2: is a partial differential equation since the function

depends on two variables and .

Order and degree of a differential equation

The order of a DE is the order of the highest ordered derivative involved in the expression . The degree of the an ordinary differential equation is the algebraic degree of its highest ordered–derivative.

Example 3: is first order, first degree ordinary DE

Example 4: is first order, first degree partial DE

Example 5: is second order, first degree ordinary DE

We say that a function is a solution to a differential equation if, when we plug it (and its various derivatives) into the equation, we find that the equation is satisfied.

Our use of the word solution has been until now somewhat informal. To be precise, we say that the continuous function is a solution of the differential equation in

on the interval I provided that the derivatives exist on I and

for all x in I. For the sake of brevity, we may say that satisfies the differential equation in on I.