Based on Euler's Method, this interactive graph illustrates a numerical method for solving differential equations. This approach is at the core of many sophisticated computer models of physical phenomena (like climate and weather).
Starting point: (x,y) = ( , )
Slope equation: dy/dx = x +
Show analytical solution:
If you know the equation for the slope of a curve (the red line for example),
and a point that the curve passes through, such as , you can integrate to find the equation of the curve:
If you don't have a starting point (initial condition), you can draw a slope field to see what the general pattern of all the possible solutions.
Even with a starting point, however, there are just times when you can't integrate the slope equation -- it's either too difficult or even impossible.
Then, what you can do is come up with an approximation of what the curve looks like by projecting along the slope from the starting point.
The program above demonstrates how it's done. This approach is called Euler's Method, and is gives a numerical approximation rather than finding the exact, analytical solution using calculus (integration).
So why use an approximation when you can find the exact solution? Because, there are quite a number of problems that are impossible or extremely difficult to solve analytically, things like: the diffusion of pollution in a lake; how changing temperature in the atmosphere gives you weather and climate; the flow of groundwater in aquifers; stresses on structural members of buildings; and the list goes on and on.
As with most types of numerical approximations, you get better results if you can reduce the step size between projections of the slope. Try changing the numbers and see.
A more detailed version, with solutions, is here: Euler's Method.
A good reference: Euler's Method by Paul Dawkins.