Slope Field Review
- Page ID
- 156414
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Standard slope field example: \(y^{\prime}=(y-3)(y+1)\), the equilibrium solution \(=\) constant solution.
\[ \notag
y=C \quad \text { if and only if } \quad y^{\prime}=0
\]
Thus to find equilibrium solution(s) if there are any, set \(y^{\prime}=0\) :
\(0=(y-3)(y+1)\) implies \(y=3\) and \(y=-1\)
Since these are constant functions, the equilibrium solutions are \(y=3\) and \(y=-1\).
If \(y^{\prime}=f(x)\) is a piecewise continuous function, the slope can only change from positive to negative and vice versa by passing through
1) a slope of 0 (horizontal tangent line) or
2) a slope of \(\infty\) (vertical tangent line) or undefined.
import matplotlib.pyplot as plt
import numpy as np
from sympy import var, plot_implicit
#Seting up the grid for us to put the arrows on
nx, ny = .5, .5
x = np.arange(-5, 5, nx)
y = np.arange(-5, 5, ny)
X, Y = np.meshgrid(x, y)
#this is the original equation
dy = Y**2-2*Y-3
dx = np.ones(dy.shape)
#normalized rates of change
dyn = dy/np.sqrt(dy**2 +dx**2)
dxn = dx/np.sqrt(dy**2 +dx**2)
plt.rcParams['figure.figsize'] = [10, 10]
plot1 = plt.plot()
plt.quiver(X, Y, dxn, dyn,
color='Blue',
headlength=5) #this is the direction field with normalized arrow size
plt.show()
A chosen point \((t_0,y_0)\) through which a solution must pass.
I.e. \((t_0,y_0)\) lies on the graph of the solution that satisfies this initial value.
A differential equation where initial value is specified.
An initial value problem can have 0 , 1 , or multiple equilibrium solutions (finite or infinite).
Long-term behavior
Suppose a solution \(y=f(t)\) to the differential equation \(y^{\prime}=(y-3)(y+1)\) passes through the point \(\left(t_0, y_0\right)\).
If \(y_0>3\), then \( \lim\limits_{t \to \infty} f(t) = \)
If \(y_0=3\), then \( \lim\limits_{t \to \infty} f(t) = \)
If \(y_0<3\), then \( \lim\limits_{t \to \infty} f(t) = \)
Standard slope field example: \(y^{\prime}=(y-3)(y+1)\)
2.5 Preview:
\(y=3\) is an unstable equilibrium solution
\(y=-1\) is a stable equilibrium solution
Note: You don't need the slope field graph to determine stability.
Note also that \(y^{\prime}=(y-3)(y+1)\) is autonomous. That is \(y^{\prime}\) depends only on \(y\) : \(\quad y^{\prime}=f(y)\)