In my earlier videos I went over using direction fields to graphically approximate solutions to differential equations, but in this video I show how we can use direction fields to numerically approximate solutions as well. This process is called the Euler's Method and is quite simply approximating the solution to an initial-value problem by using straight line segments in which the slopes are determined at each point through the differential equation. I also show in the video that as we increase the number of line segments by shortening the step size between each point, then we effectively can get closer and closer to the exact solution. With computers, this is obviously very practical since they can solve many calculations at once, and in fact many super computers are used in numerically solving differential equations using similar methods as the Euler Method. This video gives a very important illustration of a very important concept, which is numerical approximation to differential equations, so make sure to watch this video!
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Differential Equations: Euler's Method
In my earlier videos I showed how direction fields could be used to find visual or graphical approximations to solutions of differential equations.
The basic idea behind direction fields can be also used to find numerical approximations to solutions of differential equations.
To illustrate the method let's look at the same initial-value problem used to introduce direction fields:
The differential equation tells us that y'(0) = 0 + 1 = 1, so the solution curve has slope 1 at the point (0, 1).
As a first approximation to the solution we could use the linear approximation:
In other words, we could use the tangent line at (0, 1) as a rough approximation to the solution curve:
Euler's idea was to improve on this approximation by proceeding only a short distance along this tangent line and then making a midcourse correction by changing direction as indicated by the direction field.
The figure below shows what happens if we start out along the tangent line but stop when x = 0.5.
This horizontal distance of 0.5 is called the step size.
For the next line segment we can take (0.5, 1.5) as the new starting point:
This linear function is the new approximation to the solution for x > 0.5.
If we decrease the step size from 0.5 to 0.25, we get a better Euler approximation:
In general, Euler's method says to:
- Start at the point given by the initial value and proceed in the direction indicated by the direction field.
- Stop after a short time, look at the slope at the new location, and proceed in that direction.
- Keep stopping and changing direction according to the direction field.
Euler's method does not produce the exact solution to an initial-value problem, but simply gives approximations.
But by decreasing the step size (and therefore increasing the number of midcourse corrections), we obtain successively better approximations to the exact solutions.
- Compare the above figures, for example.
For the general first-order initial-value problem:
our aim is to find approximate values for the solution at equally spaced numbers x0, x1 = x0 + h, x2 = x1 + h, …, where h is the step size.
The differential equation tells us that the slope at (x0, y0) is y' = F(x0, y0), so that the approximate value of the solution when x = x1 is:
