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Section Chapter 5 Exercises
Reading Questions α―β
β Quick-Answer Questions
1. Multiple-Choice.
(a) Does time matter here?
For the autonomous equation
\begin{equation*}
y' = y^2 - 1\text{,}
\end{equation*}
the slope field shows a slope of \(-2\) at \((t,y) = (5,-1)\text{.}\) What is the slope at \((t,y) = (-10,-1)\text{?}\)
- \(0\)
- The slope doesnβt magically flatten; itβs determined by \(y\text{.}\)
- \(2\)
- The sign doesnβt flip when \(t\) changesβtime isnβt part of the slope function.
- \(-2\)
- Correct! Autonomous equations ignore \(t\) entirely. If \(y=-1\text{,}\) the slope is always \(f(-1) = -2\text{,}\) no matter the time.
- Impossible to know
- Itβs absolutely possibleβyou just evaluate \(f(y)\) at \(y=-1\text{.}\)
(b) Horizontal Shift Symmetry.
Fill in the blank: If
\(y(t)\) is a solution to an autonomous equation, then _______ is also guaranteed to be a solution.
- \(y(t + c)\)
- Correctβautonomous equations allow time-shifted versions of any solution.
- \(c \, y(t)\)
- Noβscaling the solution doesnβt necessarily work unless the equation is linear.
- \(y(t) + c\)
- Noβadding a constant changes \(y\) in a way that wonβt satisfy the DE.
- \(y(t)^c\)
- Raising the solution to a power doesnβt preserve the equationβs behavior.
(c) Time Shifts in Practice.
A biologist models population growth with the autonomous equation
\begin{equation*}
\frac{dy}{dt} = y(1 - y).
\end{equation*}
They find a solution curve \(y(t)\) that fits data collected in spring. Which of the following will always produce another valid solution?
- \(y(t + 5)\)
- Yes! Sliding the solution along the t-axis still solves the equation.
- \(2y(t)\)
- Scaling \(y\) wonβt necessarily satisfy the same DE.
- \(y(t) + 5\)
- Adding to \(y\) changes the values in a way that usually breaks the equation.
- \(y(-t)\)
- Reversing time is not the same as sliding itβit usually wonβt satisfy the DE.
(d) Equilibrium Identification.
For the equation
\begin{equation*}
\dfrac{dy}{dt} = y^2 - 4y + 3,
\end{equation*}
which of the following is an equilibrium solution?
- \(y = 1\)
-
- \(y = 2\)
-
- \(y = 3\)
-
- \(y = 4\)
-
(e) Stability Classification.
For the same equation
\begin{equation*}
\dfrac{dy}{dt} = y^2 - 4y + 3,
\end{equation*}
which of the following best describes the equilibrium at \(y = 1\text{?}\)
- Stable (sink)
-
- Unstable (source)
-
- Semi-stable (node)
-
- Not an equilibrium point
-
2. Short-Answer.
(a) πβ Phase Line Practice.
Sketch a phase line for the equation
\begin{equation*}
\dfrac{dy}{dt} = y(3 - y^2).
\end{equation*}
Then classify each equilibrium solution as a sink, source, or node.
(b) Review Questions.
-
What is an autonomous differential equation?
-
How do you find equilibrium solutions for an autonomous equation?
-
What does the phase line represent, and how is it useful?
-
How can you determine the stability of an equilibrium solution using the phase line?
Answer each question briefly:
Exercises Exercises
1. Practice Problems.
Try these problems to reinforce your understanding:
-
Sketch the slope field for
\begin{equation*}
\dfrac{dy}{dt} = y^2 - 4y\text{.}
\end{equation*}
-
Identify the equilibrium solutions and their stability.
-
Draw the phase line for the equation and describe the long-term behavior of solutions.
2. Exploring Nonlinear Dynamics.
Consider the nonlinear equation
\begin{equation*}
\dfrac{dy}{dt} = y^3 - 3y\text{.}
\end{equation*}
-
Find the equilibrium solutions.
-
Sketch the phase line and indicate the stability of each equilibrium.
-
Discuss how this equation might model a system with multiple stable states.
3. Exploring Chaos in Autonomous Systems.
Consider the equation
\begin{equation*}
\dfrac{dy}{dt} = y^2 - 2y + 1\text{.}
\end{equation*}
-
Identify the equilibrium solutions.
-
Discuss whether this system exhibits chaotic behavior or not.
-
Sketch the phase line and describe the long-term behavior of solutions.
4. Exploring Real-World Applications.
Consider a population of rabbits modeled by the equation
\begin{equation*}
\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)\text{,}
\end{equation*}
where \(r\) is the growth rate and \(K\) is the carrying capacity.
-
Identify the equilibrium solutions.
-
Sketch the phase line and describe the stability of each equilibrium.
-
Discuss how this model can help predict population dynamics over time.
5. Find the Equilibrium Solution.
Find the equilibrium solutions for the autonomous differential equation
\begin{equation*}
\dfrac{dy}{dx} = y^2\text{.}
\end{equation*}
6. Finding Equilibrium Points.
Find all equilibrium solutions of the autonomous differential equation
\begin{equation*}
\dfrac{dy}{dx} = y^2 - y^4 = 0\text{.}
\end{equation*}
7. Phase Line Sketching.
Sketch a phase line for the equation
\begin{equation*}
\dfrac{dy}{dt} = y^2 - 4y + 3.
\end{equation*}
Then classify each equilibrium point as a sink, source, or node.
You have attempted
of
activities on this page.
