CPT 03: Direction Fields

Author: John J Weber III, PhD Corresponding Textbook Section:

Expected Educational Results

Differential Equations

Separable ODEs

Definition: Separable DE

Separable DE: A DE is separable if the DE, y=f(x,y), can be written as a product y=f(x)g(y).

NOTE: The derivative y(x) may be a function of more than one variable f(x,y).

Definition: Autonomous DE

Autonomous DE: A DE is autonomous if the DE is a function of only the dependent variable, i.e., y=f(y).

NOTE: All autonomous equations are separable. The method of partial fractions is needed to solve autonomous equations.

Solutions to ODEs

Definition: A solution to an ODE are equations for curves that when substituted into the ODE results in a mathematically true statement.

Investigation 01

Let y(x)=xy2+yx.

  1. What does y(x) describe about y(x). Explain.

  2. Is y(x) a linear function? Explain.

Investigation 02

Let y(x)=xy2+yx. Find the values for y(x) at the following points (xi,yi), i.e., find f(xi,yi):

  1. (0,0)

  2. (2,1)

  3. (1,1)

  4. (12,23)

Investigation 03

From your knowledge of algebra, identify a function whose y-values never change. Explain.

Investigation 04

From your knowledge of calculus, what do you know about the derivative of the function that you identified in the previous Investigation? Explain.

Equilibrium Solutions

Definition: Equilibrium Solution

Equilibrium Solution: Equilibrium solutions are y-values, yi, such that f(yi)=0.

Stability of Equilibrium Solutions

Direction Fields

Definition: Direction Fields

Direction Fields: A direction field of a DE is a graphical representation, using vectors, of the solutions to a first-order DE.

Definition: Slope Fields

Slope Fields: A slope field of a DE is a graphical representation, using line segments, of the solutions to a first-order DE.

Definition: Phase Lines

Phase Lines: A phase line of an autonomous DE is a graphical representation of the equilibrium solutions to autonomous DEs.

Sketching Slope Fields

The key to constructing direction fields and phase lines is to understand what y describes about y by answering the following questions:

Investigation 05

Sketch the directions fields (and phase lines, where appropriate) for the following:

  1. y=y(y2)2(y4)

  2. y=y32y2

  3. y=sin(y)

  4. y=1ey

  5. y=(y1)2(y+2)2e2y

  6. y=x(y3)(1+y)

  7. y=x2y2(y4)

  8. y=xy(y+2)(y4)

xy-Plane and Phase Line

xy_plane_phase_line

Verify Slope Fields

Example: Plot the slope field for y=y(y2)2(y4)

Mathematica

 

Python

NOTE: Later in the course, we will see how to create phase planes for multivariate DEs.

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Last Modified: Sunday, 18 August 2020 21:51 EDT