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 .

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