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Differential Equations Slope Fields

  • Way of showing slope at every point of function
    • Draw table of values with desired dimensions
    • Fill in with values
    • Draw small slope lines at each coordinate
  • Calculation times can be reduced by recognizing patterns on graph

Differential Equations Slope Fields

Given y [dy/dx] + 2x = 0, find the values where [dy/dx] is undefined.
  • y [dy/dx] + 2x = 0
  • y [dy/dx] = −2x
  • [dy/dx] = [(−2x)/y]
[dy/dx] is undefined when y = 0
Solve for y given y [dy/dx] + 2x = 0
  • y [dy/dx] + 2x = 0
  • y [dy/dx] = −2x
  • ∫y  dy = −2 ∫x  dx
  • [(y2)/2] = −x2 + c
  • y2 = −2x2 + C
y = ±√{C − 2x2}
Given [dy/dx] + 2x  ey = 0, find the values where [dy/dx] is undefined.
  • [dy/dx] = −2x ey
  • 2x and ey are both continuous
  • The product of two continuous functions will also yield a continuous function
[dy/dx] is defined everywhere.
Find the x and y coordinates where [dy/dx] reaches its maximum, given [dy/dx] = [(1 − x2 + x)/(y2 + 1)]
  • [dy/dx] will reach a maximum when the numerator is as large as possible and the denominator is as small as possible.
  • We know a function will hit a relative maximum or minimum when the slope is equal to zero
  • [d/dx] (1 − x2 + x) = 0
  • −2x + 1 = 0
  • 2x = 1
  • x = [1/2]
  • Is this a minimum or a maximum? We can try test points around the critical point, or we can use the second derivative test
  • f"(x) = −2 always concave down, implies a maximum
  • [d/dy] (y2 + 1) = 0
  • 2y = 0
  • y = 0
  • f"(x) = 2 always concave up, implies a minimum
  • As a sanity check, let's make sure these values result in a positive number
  • [dy/dx] = [(1 − [1/4] + [1/2])/1] = [3/4]
[dy/dx] reaches a maximum at [[1/2], 0]
Find the conditions such that [dy/dx] = 0 if [dy/dx] = x − y
  • [dy/dx] = x − y = 0
  • x = y
[dy/dx] = 0 when x = y
Sketch the slope field of [dy/dx] = x − y
  •  x = -2 x = -1 x = 0 x = 1 x = 2
    y = -2 0 1 2 3 4
    y = -1 -1 0 1 2 3
    y = 0 -2 -1 0 1 2
    y = 1 -3 -2 -1 0 1
    y = 2 -4 -3 -2 -1 0
Sketch the slope field of [dy/dx] = x  y2
  •  x = -2 x = -1 x = 0 x = 1 x = 2
    y = -2 -8 -4 0 4 8
    y = -1 -2 -1 0 1 2
    y = 0 0 0 0 0 0
    y = 1 -2 -1 0 1 2
    y = 2 -8 -4 0 4 8
Find y given [dy/dx] = x  y2 and y(0) = 2
  • [dy/dx] = x  y2
  • [1/(y2)] [dy/dx] = x
  • [1/(y2)] dy = x  dx
  • ∫[1/(y2)] dy = ∫x  dx
  • − [1/y] = [(x2)/2] + C
  • −1 = ([(x2)/2] + C)y
  • y = [1/(−[(x2)/2] + C)]
  • y = [2/(c − x2)]
  • 2 = [2/(c − 0)]
  • c = 1
y = [2/(1 − x2)]
Find the values where [dy/dx] is undefined given [dy/dx] = [(x + 1)/(y − 1)]
  • [dy/dx] is undefined when the denominator is equal to zero
  • y − 1 = 0
  • y = 1
[dy/dx] is undefined when y = 0
Find the values where [dy/dx] is undefined or not real given [dy/dx] = [(x2)/(√{1 − y3})]
  • The function is undefined and/or not real when 1 − y3 ≤ 0. The result will not be real if you take the square root of a negative number.
  • 1 − y3 ≤ 0
  • −y3 ≤ −1
  • y3 ≥ 1
  • y ≥ 1
[dy/dx] is undefined and/or not real when y ≥ 1

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Differential Equations Slope Fields

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Slope Fields 0:08
    • What Are Slope Fields
    • Example 1
  • Example 2 6:30
  • Example 3 11:17