Differential Equations and Integration

The derivative of a function is a measurement of how it changes when it's input changes.

Another way of looking at this is that the derivative of a function is an expression for the gradient of the curve of the function. Derivatives are concerned with dependant and independent variables. You can think of the independent variable as an input to a function. It is unaffected by other variables. And the variable that is dependant is changing with the independent variable. In a derivative, the change of the dependant variable with respect to the independent variable is being expressed.

Velocity can be expressed as a derivative - the change of position with time. Here, time is the independent variable and position is the dependant variable.

A differential equation is one that relates the derivative of a function to some other function and/or another derivative.

What is known as an ordinary differential equation is a differential equation that has only one independent variable. A partial differential equation is one that has two or more independent variables.

So a differential equation that relates acceleration (dv / dt) with velocity (ds / dt) would be ordinary, since "t" is the independent variable here in both cases.

The goal of solving differential equations is to not to solve for an unknown value but to solve for an unknown function.

For example, given an unknown function for the position of a particle, S(t) and a known function for the velocity of a particle, V(t),

V(t) = d S(t) / d t

,where V(t) is the derivative of S(t) with respect to time, we can solve for the function S such that the above equation is true. Once S is found we can plug in a value of "t" to obtain a position.

We would solve this equation using a method known as Integration.

Integration is like the reverse of differentiation. The integration of a derivative will bring us back to the function we started with. This is sometimes known as anti-differentiation. The result is a expression called an indefinite integral.

More useful is to integrate over a range of values. This results in a definite integral, which is a "real value" representing the area on a graph between the curve of the function and the x-axis beween the range of values.

It is like adding up all the results (the changes) over the given range of the independent variable (e.g. between x=a and x=h for the above example). The result of adding these changes together gives us an actual value. Thus we can see how integrating velocity (the change in position with respect to time) can give us a position value. However, to get the real value, we must know and add on the initial value. When this is unknown it is acknowledged after integration by adding a constant, C.

Going back to the above example, the equation can be solved by integrating both sides with respect to "t". As shown below:

V(t) dt = S(t)

Notice that the previous right-hand-side of the equation, the derviative (d S(t) / dt), integrates to just S(t). But what is the integral of V(t) with respect to "t"? In some situations, depending on what the function V is, an exact integral is possible via analytical integration. However, in other situations numerically solving the differential equation is either more plausible or required.

That is where we come into the realm of Numerical Integration.

Research on numerical integration coming soon.


Sources:

http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations
http://www.physics.ohio-state.edu/~physedu/mapletutorial/tutorials/diff_eqs/intro.html
http://www.myphysicslab.com/what_diff_eq.html
http://en.wikipedia.org/wiki/Integral
http://en.wikipedia.org/wiki/Differential_equation
http://en.wikipedia.org/wiki/Derivative