CPT 03: Direction Fields

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

• Section 1.3 – Direction Fields

Expected Educational Results

• Objective 3–1: I can distinguish between autonomous equations and non-autonomous equations.

• Objective 3–2: I can identify equilibrium solutions to ordinary differential equations.

• Objective 3–3: I can identify stability of equilibrium solutions to ordinary differential equations.

• Objective 3–4: I can sketch typical solution(s) to ordinary differential equations.

• Objective 3–5: I can use technology to sketch direction fields to ordinary differential equations.

Differential Equations

Separable ODEs

Definition: Separable DE

Separable DE: A DE is separable if the DE, $y^{\prime }=f(x,y)$$y^{\,\prime}=f(x,y)$, can be written as a product $y^{\prime }=f(x)g(y)$$y^{\,\prime}=f(x)g(y)$.

NOTE: The derivative $y^{\prime }(x)$$y^{\,\prime}(x)$ may be a function of more than one variable $f(x,y)$$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^{\prime }=f(y)$$y^{\,\prime}=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^{\prime }(x)=x{y}^2+y-x$$y^{\,\prime}(x)=xy^2+y-x$.

1. What does $y^{\prime }(x)$$y^{\,\prime}(x)$ describe about $y(x)$$y(x)$. Explain.

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

Investigation 02

Let $y^{\prime }(x)=x{y}^2+y-x$$y^{\,\prime}(x)=xy^2+y-x$. Find the values for $y^{\prime }(x)$$y^{\,\prime}(x)$ at the following points $(x_i,y_i)$$(x_i,y_i)$, i.e., find $f(x_i,y_i)$$f(x_i,y_i)$:

1. $(0,0)$$(0,0)$

2. $(-2,1)$$(-2,1)$

3. $(-1,1)$$(-1,1)$

4. $(-\frac{1}{2},-\frac{2}{3})$$\left(-\frac{1}{2},-\frac{2}{3}\right)$

Investigation 03

From your knowledge of algebra, identify a function whose $y$$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$$y$-values, $y_i$$y_i$, such that $f(y_i)=0$$f(y_i)=0$.

Stability of Equilibrium Solutions

• An equilibrium solution is stable if and only if solutions near them converge to $y=y_i$$y=y_i$.

• An equilibrium solution is semi-stable if:

  • solutions near $y=y_i$$y=y_i$ converge to $y=y_i$$y=y_i$ for $y>y_i$$y>y_i$ and diverge for $y<y_i$$y<y_i$, OR

  • solutions near $y=y_i$$y=y_i$ converge to $y=y_i$$y=y_i$ for $y<y_i$$y<y_i$ and diverge for $y>y_i$$y>y_i$.

• An equilibrium solution is unstable if and only if solutions are neither stable solutions nor semi-stable 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^{\prime }$$y^{\,\prime}$ describes about $y$$y$ by answering the following questions:

• Is the DE autonomous? Explain.

• Identify the equilibrium solutions to the DE. Explain.

• Classify the equilibrium solutions as stable, semi-stable, or unstable. Explain.

• Sketch several solutions to completely describe solutions to the DE. Explain.

Investigation 05

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

1. ${\displaystyle y^{\prime }=y(y-2{)}^2(y-4)}$$\displaystyle y^{\,\prime}=y(y-2)^2(y-4)$

2. ${\displaystyle y^{\prime }=y^3-2y^2}$$\displaystyle y^{\,\prime}=y^3-2y^2$

3. ${\displaystyle y^{\prime }=\sin (y)}$$\displaystyle y^{\,\prime}=\sin{(y)}$

4. ${\displaystyle y^{\prime }=1-e^y}$$\displaystyle y^{\,\prime}=1-e^y$

5. ${\displaystyle y^{\prime }=(y-1{)}^2(y+2{)}^2{e}^{-2y}}$$\displaystyle y^{\,\prime}=(y-1)^2(y+2)^2e^{-2y}$

6. ${\displaystyle y^{\prime }=x(y-3)(1+y)}$$\displaystyle y^{\,\prime}=x(y-3)(1+y)$

7. ${\displaystyle y^{\prime }=x^2{y}^2(y-4)}$$\displaystyle y^{\,\prime}=x^2y^2(y-4)$

8. ${\displaystyle y^{\prime }=xy(y+2)(y-4)}$$\displaystyle y^{\,\prime}=xy(y+2)(y-4)$

$xy$$xy$-Plane and Phase Line

Verify Slope Fields

Example: Plot the slope field for $y^{\prime }=y(y-2{)}^2(y-4)$$y^{\prime}=y(y-2)^2(y-4)$

Mathematica

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(* Example, plot the slope field for y'=y(y-2)^2(y-4) *)
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(* Plot the direction field *)
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VectorPlot[{1, y (y - 2)^2 (y - 4)}, {x, -10, 10}, {y, -10, 10}]
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(* Typically, StreamPlot is easier to use *)
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StreamPlot[{1, y (y - 2)^2 (y - 4)}, {x, -10, 10}, {y, -10, 10}]
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(* NOTE: the range of values for x and y can be changed as needed *)  

Python

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# Import the needed packages
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import numpy as np
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import matplotlib.pyplot as plt
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#
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# Set up the mesh grid; change as needed
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Y, X = np.mgrid[-3:7:100j, -5:5:100j]
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#
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# Set up the autonomous DE as parametric equations
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# Change V as needed ** is exponentiation
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U = X
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V = Y*(Y-2)**2*(Y-4)
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#
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# Set up streamplot and plot
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fig, ax = plt.subplots()
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ax.streamplot(X, Y, U, V)
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#
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# Show the plot
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plt.show()

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

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

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