The Euler Methods

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Section 9.2

• The Euler Methods

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EULERS METHOD
One of the simplest techniques for approximating
solutions of differential equations is Eulers
Method, which is also known as the method of
tangent lines.
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PROCEDURE FOR EULERS METHOD
1. Given y' f (x, y), y(x0) y0, find the
slope of the tangent line at (x0, y0). The slope
is y'(x0, y0)  f  (x0, y0). We denote this
slope by y'0. 2. Find a point (x1, y1) (x0 h,
y1) on the tangent line by the formula y1 y0
hy'0. The variable h represents the step size
which is reasonably small. 3. Using the same
value for h, we find the slope y'1 at (x1, y1).
Find (x2, y2) by x2 x1 h and
y2  y1  hy'1.
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PROCEDURE (CONCLUDED)
4. By continuing in the above manner, we are
able to draw an approximate solution
curve. NOTE In general, where xn x0 nh.
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ERROR
Absolute Error Relative Error Percentage
Relative Error
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IMPROVED EULERS METHOD
This method is the same as Eulers method except
that it uses a more accurate approximations. It
uses the improved Eulers formula, or Heuns
formula.