Chapter 8

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=y\left(y^{2}-1\right) $$

\text { In Problems 1-8, solve each pure-time differential equation. } \frac{d y}{d t}=t+\sin t, \text { where } y(0)=0

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d t}+\frac{y}{t}=\frac{1}{t^{2}} $$

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=y^{3}+y $$

\text { In Problems , solve each pure-time differential equation. } \frac{d y}{d t}=e^{-3 t}, \text { where } y(0)=10 .

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d t}+\frac{3 y}{t}=t $$

Find the equilibria of the following differential equations. $$ \frac{d x}{d t}=x^{2}-3 x+2 $$

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d y}{d x}=\frac{1}{x}, \text { where } y(1)=0 . $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d t}-y=t+1 $$

Find the equilibria of the following differential equations. $$ \frac{d x}{d t}=6+5 x+x^{2} $$

Suppose that a tank holds 1000 liters of water, and \(2 \mathrm{~kg}\) of salt is poured into the tank. (a) Compute the concentration of salt in \(\mathrm{g}\) liter \(^{-1}\). (b) Assume now that you want to reduce the salt concentration. One method would be to remove a certain amount of the salt water from the tank and then replace it by pure water. How much salt water do you have to replace by pure water to obtain a salt concentration of \(1 \mathrm{~g}\) liter \(^{-1} ?\) (c) Another method for reducing the salt concentration would be to hook up an overflow pipe and pump pure water into the tank. That way, the salt concentration would be gradually reduced. Assume that you have the choice of two pumps, one that pumps water at a rate of 1 liter \(\mathrm{s}^{-1}\), the other at a rate of 2 liter \(\mathrm{s}^{-1}\). For each pump, find out how long it would take to reduce the salt concentration from the original concentration to 1 gliter \(^{-1}\). (Note that the rate at which water enters the tank is equal to the rate at which water leaves the tank.) (d) Show that, whichever pump you use in part (c), you need more pure water if you use the pump method than if you follow the method in (b). Can you explain why?

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d y}{d x}=\frac{1}{1-x^{2}}, \text { where } y(0)=0 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d t}+y=t^{2} $$

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=\frac{y-2}{y+1} $$

A cell constantly gains or loses small molecules to its environment because the small molecules are able to diffuse through the cell membrane. We will build a model for this process. Suppose a molecule is present in the cell at a concentration \(C(t)\), and present in its environment at a concentration \(C_{\infty}\) (you may assume \(C_{\infty}\) is a constant). One model for the diffusion of molecules across the cell membrane is that the rate at which molecules travel through the membrane is proportional to the difference in concentration between the cell and its surroundings. That is: Rate at which $$ \text { molecules flow out }=k\left(C-C_{\infty}\right) $$ of cell The constant \(k\) is known as the permeability of the membrane: \(k>0\), and \(k\) depends on the surface area of the cell and the chemistry of the membrane, as well as the type of molecule. (a) Starting with a word equation for the amount of small molecules in the cell, show, if the cell volume is \(V\), then: $$ \frac{d C}{d t}=-\frac{k}{V}\left(C-C_{\infty}\right) $$ (b) Find the equilibrium of \((8.53)\) and use a graphical analysis to determine whether it is stable or unstable. (c) Suppose that the molecule we are studying is produced within the cell. The cell produces the molecule at a rate \(r\); that is, a quantity \(r\) is produced (added to the cell) in unit time. Explain why the differential equation for the concentration of molecules in the cell should be modified to: $$ \frac{d C}{d t}=-\frac{k}{V}\left(C-C_{\infty}\right)+\frac{r}{V} $$ (d) Analyze Equation (8.54) to find the equilibrium value of the cell concentration. Is this equilibrium stable or unstable? You may use a graphical argument or calculate the eigenvalue to determine the equilibrium's stability.

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d x}{d t}=\frac{1}{1-t}, \text { where } x(0)=2 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d x}+\frac{y}{x+2}=x-1 $$

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=\frac{y-1}{y^{2}+1} $$

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d x}{d t}=\sin (2 \pi(t+3)), \text { where } x(3)=1 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d x}+\frac{y}{x+2}=x $$

Find the equilibria of the following differential equations. $$ \frac{d x}{d t}=x^{8}-1 $$

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d s}{d t}=\sqrt{t+1}, \text { where } s(0)=1 $$

$$ \frac{d y}{d x}-\frac{y}{x(x+1)}=1 $$

Find the equilibria of the following differential equations. $$ \frac{d y}{d t}=y^{1 / 3}-1 $$

Insulin pumps treat patients with type I diabetes by releasing insulin continuously into the fat in the patient's stomach or thigh. We will develop a model for the transport of insulin from the site where it is released by the pump, by treating the fat as a compartment in a single-compartment model. Let's suppose that the pump releases insulin at a constant rate, \(r(r\) is the amount added in one unit of time). (a) Explain why, if insulin is not transported from the site of release, the amount of insulin at the site of release, \(a(t)\), will obey a differential equation: $$ \frac{d a}{d t}=r $$ (b) From the fat, the insulin enters the patient's bloodstream. Suppose that a fraction \(p\) of the insulin present in the patient's fat enters the blood in unit time. Explain why: $$ \frac{d a}{d t}=r-p a $$ (c) Find the equilibrium from the differential equation in part (b) and determine whether this equilibrium is stable or unstable.

\text { In Problems , solve each pure-time differential equation. }$$ \frac{d h}{d t}=4-16 t^{2}, \text { where } h(1)=0 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d x}+\frac{2(x+1) y}{x(x+2)}=x $$

Find the equilibria of the following differential equations. $$ \frac{d N}{d t}=N e^{-N} $$

Compartment models are used to model the flow of traffic between different roads, by treating each road as a compartment. As an example, consider how the number of cars on a freeway on-ramp, \(N(t)\), changes with time. For a simplified model let's assume that cars join the on-ramp at a constant rate \(q\) (that is, \(q\) cars join the on-ramp in one unit of time). Cars then leave the on-ramp by entering the freeway itself. Assume that a fraction \(f\) of the cars on the on-ramp enter the freeway in one unit of time. (a) Derive a differential equation for \(N(t) .\) Your differential equation will include the unknown constants \(f\) and \(q\). (b) Analyze your model from part (a) to find the equilibrium number of cars on the on-ramp, and determine whether this equilibrium is stable or unstable. (c) Suppose that the maximum capacity of the on-ramp is 90 cars, and the rate at which cars flow onto the on-ramp is \(q=60\) cars per min. Find the value of \(f\) that is needed to keep \(N\) below the on-ramp's capacity.

Suppose that the volume \(V(t)\) of a cell at time \(t\) changes according to $$ \frac{d V}{d t}=1+\cos t \quad \text { with } V(0)=5 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d x}{d t}+(t-1) x=t-1 $$

Find the equilibria of the following differential equations. $$ \frac{d N}{d t}=N \ln N, N>0 $$

In our compartment model we assumed that inflows and outflows are matched at \(q\) to keep the volume of water in the tank constant. It's often useful when modeling, for example, the flow of pollutant into a pristine environment, to consider what can occur if the inflows and outflows do not match. Let's assume that the tank initially contains a volume \(V_{0}\) of water. Water flows into the tank at rate \(q_{\mathrm{in}}\), and out of the tank at rate \(q_{\text {out. }}\) (You may assume \(q_{\text {in }}>q_{\text {out } .}\) ) Suppose that the water flowing into the tank contains a concentration \(C_{I}\) of solute. As usual we write \(C(t)\) for the concentration in the tank. (a) Show that the concentration in the tank can be modeled using a differential equation: $$ \frac{d}{d t}(C V)=q_{\text {in }} C_{I}-q_{\text {out }} C $$ (b) Previously we were able to treat \(V\) as a constant. Now \(V\) changes with time. Derive a formula for \(V(t)\). (c) By substituting your formula for \(V(t)\) into (a), derive a differential equation for \(C(t)\). (d) In general we cannot analyze the behavior of the solution \(C(t)\) using techniques from Section \(8.2 .\) Why not? (e) Let's assume \(C_{l}=0\). Then show that your equation from (c) can be written as: $$ \frac{d C}{d t}=\frac{-q_{\mathrm{in}} C}{V_{0}+\left(q_{\mathrm{in}}-q_{\mathrm{out}}\right) t} $$ (f) Assume some definite values for the constants in \((8.57):\) \(q_{\mathrm{in}}=2, q_{\text {out }}=1\), and \(V_{0}=20 .\) Assuming \(C(0)=1\), solve \((8.57)\) to find \(C(t) .\) Show that \(\lim _{t \rightarrow \infty} C(t)=0\).

Suppose that the amount of phosphorus in a lake at time \(t\). denoted by \(P(t)\), follows the equation $$ \frac{d P}{d t}=3 t+1 \quad \text { with } P(0)=0 $$

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d x}{d t}+\frac{t x}{t^{2}+1}=t $$

Find the equilibria of the following differential equations. $$ \frac{d N}{d t}=\sin N $$

Denote by \(p=p(t)\) the fraction of occupied sites in the patchy habitat model, and assume that $$ \frac{d p}{d t}=2 p(1-p)-p \quad \text { for } t \geq 0 $$ (a) Set \(g(p)=2 p(1-p)-p .\) Graph \(g(p)\) for \(p \in[0,1]\). (b) Find all equilibria of \((8.58)\) that are in \([0,1] .\) Use your graph from (a) to determine their stability. (c) Now use the eigenvalue approach to analyze the stability of the equilibria that you found in (b).

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d x}+\frac{y}{x}=y $$

Find the equilibria of the following differential equations. $$ \frac{d N}{d t}=N \cos 2 N $$

Denote by \(p=p(t)\) the fraction of occupied sites in the patchy habitat model, and assume that $$ \frac{d p}{d t}=0.5 p(1-p)-1.5 p \quad \text { for } t \geq 0 $$ (a) Set \(g(p)=0.5 p(1-p)-1.5 p .\) Graph \(g(p)\) for \(p \in[0,1]\). (b) Find all equilibria of \((8.59)\) that are in \([0,1]\). Use your graph in (a) to determine their stability. (c) Use the eigenvalue approach to analyze the stability of the equilibria that you found in (b).

. Drug Absorption Drug enters a patient's blood by being absorbed from the gut. Assume that the drug enters the patient's blood at a rate that depends on time as \(c e^{-r t}\) where \(c\) and \(r\) are positive constants (the rationale for this formula will be discussed in Section \(8.4\) ) and the drug is eliminated at constant rate \(k\). So: $$ \frac{d M}{d t}=c e^{-r t}-k $$ (a) Assuming \(c>k\) and \(M(0)=0\) (there is no drug present in the patient's blood at the start of the experiment), solve this differential equation. (b) Suppose \(k>0\). What does your solution predict will happen to \(M(t)\) as \(t \rightarrow \infty\) ? Does your answer make sense? (In reality, drug can only be removed at a constant rate until all drug is removed from the blood. That is, the rate of elimination will be \(k\) if \(M>0\) and 0 once \(M\) drops to \(0 .)\) (c) Assume \(k=0\) (i.e., this drug is never eliminated from blood, or is eliminated so slowly that elimination can be neglected). Show that \(\lim _{t \rightarrow \infty} M(t)=\frac{c}{r}\).

Find the general solution of the differential equations in Problems 1-12 using the method of integrating factors: $$ \frac{d y}{d x}+x=y $$

For Problems 13-28 make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=y-1 $$

Subpopulation Interactions in Patchy Habitats To derive our model for patchy habitat we assumed that a fixed fraction, \(m\), of occupied sites became extinct in each unit of time. Often, however the survival of the population at a site depends on the number of subpopulations in the surrounding sites. If different subpopulations compete for limited resources, then the per site mortality rate may not be a constant, but may increase with \(p\) because, as \(p\) increases, competition between subpopulations increases. In questions 13 and 14 we will study the effect of different models for competition between subpopulations. The term \(p^{2}\) describes the density-dependent extinction of patches; that is, the per-patch extinction rate is \(p\), and a fraction \(p\) of patches are occupied, resulting in patches going extinct at a total rate of \(p^{2}\). The colonization of vacant patches is the same as in the Levins model. Then the fraction of occupied patches obeys a differential equation: $$ \frac{d p}{d t}=c p(1-p)-p^{2} $$ where \(c>0\). (a) Show that there are two possible equilibrium values for \(p\) in \([0,1]\) (which you should calculate) and determine their stability. (b) Does the patch model always predict a nontrivial equilibrium when \(c>0\) ? Contrast with what we found for the Levins model in Section 8.3.2.

In Problems 13-18, solve each autonomous differential equc tion. $$ \frac{d y}{d t}=2 y, \text { where } y(0)=2 $$

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=\frac{y}{t}-t^{2} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=2-y $$

In Problems , solve each autonomous differential equc tion. $$ \frac{d y}{d t}=2(1-y), \text { where } y(0)=0 $$

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d t}=\frac{y}{t+1} $$

For make vector field plots of each of the differential equations. Find any equilibria of each differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. $$ \frac{d y}{d t}=4-y^{2} $$