As we have already seen, we may not be able to attain a solution of a differential equation easily, but rather than drawing a slope field we may desire to obtain numerical estimates for solutions to differential equations instead.
Euler’s Method is a numerical method that uses the idea of tangent lines for a short distance to approximate the solution to an initial-value problem.
Now, from our previous study, we know that the basic idea behind Slope Fields, or Directional Fields, is to find a numerical approximation to a solution of a Differential Equation, but Euler’s Method is a technique where we create a table rather than a graph and can be incredibly accurate.
Table for finding Euler’s Approximation
Here’s the big idea…if we zoom in small enough, every curve looks like a straight line, and therefore, the tangent line is a great way for us to calculate what is happening over a period of time.
With this idea, we embark on our study of Euler’s Method. As SOS Math Math nicely states, with this idea that, close to a point, a function and its tangent line do not differ very much, we will obtain numerical approximations to a solution.
We will begin by learning how to create an easy table, for which we will be able to solve any and all questions dealing with Euler’s Method and utilize this table for several examples.
Then we will see how to fill-in, or complete, the table starting with the initial value and proceeding in the direction indicated by the direction field.
Lastly, we will then look a question where we compare our three techniques for Differential Equations: Slope Fields, Euler’s Method, and finding particular solutions via Separable Differential Equations.
Eulers Method Video
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