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Slope Fields and Euler’s Method

Learn about slope fields, initial value problems, and Euler’s method in differential equations. Understand how to solve initial value problems, draw slope fields, and apply Euler’s method for approximating solutions.

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Slope Fields and Euler’s Method

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  1. Slope Fields and Euler’s Method

  2. Given: find • When taking an antiderivative that is not dealing with a definite integral, be sure to add the constant at the end. We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

  3. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation. If we have some more information we can find C. Given: y’ = 2x and f(1) = 4, find the equation for y.

  4. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but recent AP tests have asked students to draw a simple one by hand. Initial value problems and differential equations can be illustrated with a slope field.

  5. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 0 0 0 0 1 0 0 2 0 0 3 0 2 1 0 1 1 2 2 0 4 -1 -2 0 0 -4 -2

  6. If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

  7. When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.

  8. Euler’s Method Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind. (When this portrait was made he had already lost most of the sight in his right eye.) Leonhard Euler 1707 - 1783

  9. (function notation) (base of natural log) (pi) (summation) (finite change) It was Euler who originated the following notations: Leonhard Euler 1707 - 1783

  10. We will practice with an easy one that can be solved. Initial value: There are many differential equations that can not be solved. We can still find an approximate solution.

  11. Steps for using Euler’s Method: • Use the d.e. to find the slope dy/dx at that point. • Begin a the point specified by the initial condition. • Increase x by a small amount Δx. Increase y by a small amount Δy where Δy = (dydx)Δx. This defines a new point (x + Δx, y + Δy) that lies along the linearization. • Use this new point, return to step 2. Repeating the process constructs the graph to the right of the initial point.

  12. Exact Solution:

  13. It is more accurate if a smaller value is used for dx. It gets less accurate as you move away from the initial value. Examples 9 and 10 also show Euler’s Method I also have a program that will do an Euler Table and graph.

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