Direction (Slope) Fields and Euler

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Lesson 9-2

• Direction (Slope) FieldsandEulers Method

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Solutions to Differential Equations

• The solution to a differential equation is a
  function and it may be general or particular
  (given an initial condition)
• There are 3 ways to solve a differential
  equation
• Analytically (Separate Integrate)
• Graphically (with Slope Fields)
• Numerically (with Eulers Method)

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Direction or Slope Field
The directional field allows us to visualize the
general shape of the solution curves by
indicating the direction (the slope at that
point) in which the curves proceed at each
point. We draw the curve, given an initial
condition, so that it is parallel to the nearby
line segments (slopes in the field)
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Slope Field Example
f(0) -2
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Example 1

• Given
• dy -xy²
• ---- -------- with f(-1) 2
• dx 2
• Graph the slope field at the twelve indicated
  points

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Example 2

• Given
• dy -2x
• ---- -------- with f(1) -1
• dx y
• Graph the slope field at the twelve indicated
  points

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Example 3

• Given
• dy
• ---- x4(y 2) with f(0) 0
• dx
• Graph the slope field at the twelve indicated
  points

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Example 4

• Given
• dy
• ---- x2(y 2) with f(0) 3
• dx
• Graph the slope field at the twelve indicated
  points

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Eulers Method
Eulers method is an iterative process (like
fractals in Geometry) in which the next value is
dependent on the previous value. It is used in
many different applications and is the foundation
of an area of mathematics called Time Series
Analysis. Some hurricane models are based on
these same type of algorithms. For the first
estimate y1 y0 h F(x0, y0) (Stewarts
notation h is ?x and F is slope) y1 y0
slope(x0,y0) ?x (from the notation in your
notes) For the nth estimate yn yn-1 h
F(xn-1, yn-1) yn yn-1 slope(xn-1,yn-1)
?x These iterative processes are just what
spreadsheets were made for. The next slide shows
example 3 on page 597 of Stewart
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Example 3 page 597
i slope x y ?x 0.1
0 1 0 1    
1 1.2 0.1 1.1    
2 1.42 0.2 1.22    
3 1.662 0.3 1.362    
4 1.9282 0.4 1.5282    
5 2.22102 0.5 1.72102    
6 2.543122 0.6 1.943122    
7 2.8974342 0.7 2.197434    
8 3.28717762 0.8 2.487178    
9 3.715895382 0.9 2.815895    
10 4.18748492 1 3.187485    
x y xi ?x yi-1 ?xslopei-1 yi-1 ?xslopei-1
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Euler Example 1
dy x ----- ------ dx y²
Given with y(0) 1 Use
Eulers method starting at x 0 with a step
size of 0.2 (?x) to approximate y(1). yn yn-1
slope(xn-1,yn-1) ?x
i slope x y ?x 0.2
0 0 0 1
1 0.2 0.2 1
2 0.369822 0.4 1.04
3 0.483513 0.6 1.113964
4 0.545809 0.8 1.210667
5 0.57407 1 1.319829
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Euler Example 1 cont
Now find the particular solution to
and find y(1). Compare the exact and
approximate values. Remember the steps to solve
simple differential equations!
dy x ----- ------ dx y²
dy x ----- ------ dx y²
y² dy x dx
? y² dy ? x dx
?y³ ½x² C y (3/2)x²
C y(0) 1 so C 1 y(1) (3/2) 1
1.3572 vs 1.319829
Important to note that the C is underthe cube
root!!
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with ?x 0.1 EulersMethod 1.339315
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Summary Homework

• Summary
• Slope field is just the slope of the function,
  the value of the derivative graphed at the point
• Eulers method, an iterative method, allows us to
  solve for the function analytically
• Homework
• pg 599 601 3-7, 11, 12, (Euler Problem 23)