5. Euler’s Method

Euler (oiler) 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

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

We are still looking at differential equations that can not be solved. Yesterday we saw that slope fields gave us a graphical solution, Euler came up with a numeric method based on tangent line approximations Euler’s method basically involves “walking out along a tightrope” from an initial point along its tangent line. Instead of walking along the same line the whole time (as in a tangent line approximation), we change tangent lines with each step (of length ∆x). This involves recalculating the point and slope after each step. This will produce a much more accurate approximation than simply using the original tangent line.

The error gets worse as you get further away from initial value The error gets better if you use a smaller ∆ x If the curve is concave down, Euler overestimates the y value, if the curve is concave up Euler underestimates it

Euler’s Method Table (you need to memorize!!) (x,y) ∆x (x + ∆x, y + ∆y)

Example 1 Use Euler’s Method with two equal steps for dy/dx= x – 2 and if y=5 when x=0, to approximate y(0.8) (x,y) ∆x (x + ∆x, y + ∆y)

Example 2 Use Euler’s Method for dy/dx= 2x – y and f(2) = 3 with five equal steps to approximate f(1.5). (x,y) ∆x (x + ∆x, y + ∆y)

Example 3 Assume that f and f’ have the values given in the table. Use Euler’s Method with two equal steps to approximate the value of f(2.6)