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Unformatted text preview: 6.1 day 2 Eulers Method Leonhard Euler 1707  1783 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.) Greg Kelly, Hanford High School, Richland, Washington Leonhard Euler 1707  1783 It was Euler who originated the following notations: e (base of natural log) ( 29 f x (function notation) (pi) i ( 29 1(summation) y (finite change) There are many differential equations that can not be solved. We can still find an approximate solution. We will practice with an easy one that can be solved. 2 dy x dx = Initial value: 1 y = 2 dy x dx = 1 y = n n x n y dy dx dy 1 n y + 0.5 dx = 1 1 1 .5 1 1 .5 1.5 2 1 1.5 2 1 2.5 dy dx dy dx = 1 n n y dy y + + = 3 1.5 2.5 3 1.5 4.0 4 2.0 4.0 dy dx dy dx = 1 n n y dy y + + = 2 dy x dx = n n x n y dy dx dy 1 n y + 0.5 dx = 1 1 1 .5 1 1 .5 1.5 2 1 1.5 2 1 2.5 1 y = 2 dy x dx = ( 29 0,1 0.5 dx = 2 dy x dx = 2 y x C = + 1 C = + 2 1 y x = + Exact Solution: It is more accurate if a smaller...
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This note was uploaded on 02/12/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith
 Math

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