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Reading: $\S$ 2.5: pp 58-63 (Intro and Example 1. Problem 5 of $\S 2.5$ )
1. A. Autonomous first order equations
- Each problem below lists an autonomous equation of the form $\frac{d y}{d t}=f(y)$ for a function $y(t)$.
For each:
(i) Sketch the graph of $f(y)$ vs $y$. Below or next to this graph, draw the phaseline.
(ii) Determine the equilibrium points and classify each of them as asymptotically stable, unstable or semi-stable.
(iii) Sketch several solution curves in the $t-y$ plane.
(a) $\frac{d y}{d t}=y$
(b) $\frac{d y}{d t}=y^{2}$ (c) $\frac{d y}{d t}=k-r y, k>0, r>0$
(d) $(*) \frac{d y}{d t}=k+r y, k>0, r>0$ (e) $\frac{d y}{d t}=y(y-1)(y-2)$
(f) $\frac{d y}{d t}=-k(1-y)^{2}, k>0$ (g) $\frac{d y}{d t}=y^{2}\left(1-y^{2}\right)$
- §1.1:11-16. Show your work justifying your answer! 3. In each of the following problems, write down a differential equation of the form $d y / d t=a y+b$ whose solutions have the required behaviour as $t \rightarrow \infty$. (You need to show your work.)
(a) $\left(^{*}\right)$ All solutions approach $y=2$ :
(b) All solutions approach $y=-1$ :
(c) All solutions diverge from $y=3$ :
(d) $y=1 / 3$ is a steady solution:
2. B. Applications from physics
- $(*)$ Suppose a given spherical raindrop evaporates at the rate
$$
\frac{d V}{d t}=-2 V^{2 / 3}
$$
where $V \geq 0$ is its volume, and the initial volume is $V(0)=8$.
(a) Sketch the graph of $f(V)$ vs $V$. Draw the phaseline.
(b) Find the solution $V(t)$. Sketch the solution in the $t$ – $V$ plane. Is this solution consistent with your result in (a)? (c) At what time has the drop completely evaporated?
(d) State two solutions to the differential equation that satisfy $V(3)=0$. Show the graph of both solutions in your plot in (b).
(e) You have clearly demonstrated non-uniqueness for the initial value problem in (d). Why does this not contradict theorem 2.4.2? 5. Consider an object of mass $m>0$, moving in the vertical direction with velocity $v$, where positive velocity is taken to correspond to upward motion. The gravitational acceleration is $g>0$. Friction is assumed to be proportional to velocity, with constant of proportionality $\gamma>0$. Newton’s second law yields the autonomous equation $d v / d t=f(v)$ for the velocity $v(t)$
$$
m \frac{d v}{d t}=-m g-\gamma v
$$
(a) Sketch the graph of $f(v)$ vs $v$. Below or next to this graph, draw the phase line. Next to it, sketch several solution curves in the $t-v$ plane.
(b) Using only your graphs in (a), what is the terminal velocity of the object, assuming it never hits ground? (c) This equation is linear (so the method of integrating factors applies), and separable. Find the general solution using a method of your choice.
(d) What is the terminal velocity of the object, based on your answer in (c)? Is your answer consistent with your answer in (b)? 6. Based upon observations, Carla developed the differential equation $\frac{d T}{d t}=-0.08(T-72)$ to predict the temperature in her vanilla chai tea. In the equation, $T$ represents the temperature of the chai in ${ }^{\circ} \mathrm{F}$ and $t$ is time in minutes. Carla has a cup of chai whose initial temperature is $110^{\circ} \mathrm{F}$ and her friend Peter has a cup of chai whose initial temperature is $120^{\circ} \mathrm{F}$. According to Carla’s model, will there be a point in time when the two cups of chai have exactly the same temperature?
(a) Yes
(b) No
(c) Can’t tell with the information given
Briefly justify your answer.
- Consider the simple model for population growth
$$
\frac{d P}{d t}=r P, \quad r>0
$$
(a) Draw the phase line and several solution curves in the $t-P$ plane. (b) Mark all the statements below that are correct
The dimensions of $r$ are $[r]=\frac{1}{\text { time }}$
This is an example of a nonlinear differential equation.
Solutions to this equation for any initial condition exist for all times.
The model is a good model for populations whose growth is not limited by space and food resources, or interaction with other species.
The model is a good model when the population size is near the capacity of the environment, with limited amount of food and space
The model predicts that any nonzero populations will grow to $\infty$ as $t$ approaches some finite critical time $t \rightarrow t_{c}$.
The model predicts that any nonzero populations will grow to $\infty$ as $t \rightarrow \infty$.
The solution that satisfies the initial condition $P\left(t_{1}\right)=P_{1}$ is $P(t)=P_{1} e^{r\left(t-t_{1}\right)}$.
(c) Suppose $P\left(t_{1}\right)=P_{1}$. Let $\tau=t-t_{1}$ (the time elapsed since $t_{1}$ ). Then
- $P(t)=2 P_{1}$ when $\tau=$
- $P(t)=8 P_{1}$ when $\tau=$
- $P(t)=1024 P_{1}$ when $\tau=$
- A scientist develops the logistic population model $P^{\prime}=0.2 P\left(1-\frac{P}{8}\right)$ to describe her research data. This model has an equilibrium value of $P=8$. In this model, from the initial condition $P_{0}=4$, the population will never reach the equilibrium value because:
(a) the population will grow toward infinity
(b) the population is asymptotically approaching a value of 8 .
(c) the population will drop to 0
Briefly describe how you determined your answer. 9. Consider the logistic population growth given by the autonomous equation $\frac{d P}{d t}=r P\left(1-\frac{P}{\kappa}\right)$. where $r, \kappa>0$.
(a) Draw the phaseline, classify the equilibria and sketch several graphs of solutions in the $P-t$ plane. (b) Find the solution if $0<P(0)<\kappa$ and check that the limiting behaviour agrees with your sketch in (a).
(c) Find the solution if $P(0)=\kappa$. 10. A population $P(t)$ of field mice, where $P$ is the number of mice and $t$ is measured in months from some starting point, grows at the rate of $50 \%$ per month. However, owls in the neighbourhood eat them at the rate of $15 /$ day (assume that there are 30 days/month).
(a) Write down a differential equation for $P(t)$.
(b) Draw the phaseline and a few integral curves.
(c) Describe how the evolution of the mouse population in time depends on the initial population $P_{0}=P(0)$.
(d) Find the time at which the population becomes extinct if $P_{0}=850$.
- Consider the differential equation $\frac{d y}{d x}=3 x y+y^{2}+1$ with initial condition $y(2)=6$.
(a) Theorem 2.4.2 guarantees that a unique solution $y(x)$ to this initial value problem exists. Find the equation of the tangent line to $y(x)$ at $(2,6)$.
(b) Without solving the problem, what can you say about the interval in which the solution is guaranteed to exist?
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Applied Ordinary Differential Equations
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