Calc06_6 - 6.6 Euler's Method Leonhard Euler made a huge...

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6.6 Euler’s 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
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Leonhard Euler 1707 - 1783 It was Euler who originated the following notations: e (base of natural log) ( 29 f x (function notation) π (pi) i ( 29 1 - (summation) y (finite change)
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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: 0 1 y =
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2 dy x dx = 0 1 y = n n x n y dy dx dy 1 n y + 0.5 dx = 0 0 1 0 0 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 + + =
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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 = 0 0 1 0 0 1 1 .5 1 1 .5 1.5 2 1 1.5 2 1 2.5 0 1 y =
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2 dy x dx = ( 29 0,1 0.5 dx = 2 dy x dx = 2 y x C = + 1 0 C = + 2 1 y x = + Exact Solution:
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It is more accurate if a smaller value is used for dx .
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