Chapter 11

Use the mass action law to translate each chemical reaction into a system of differential equations. \(\mathrm{A}+\mathrm{B} \underset{k-}{\stackrel{k_{+}}{\rightleftarrows}} \mathrm{C}\)

In Problems 19-26, solve the given initial-value problem. \(\left[\begin{array}{c}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{rr}-3 & 0 \\ 4 & 2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=-5\) and \(x_{2}(0)=5\).

Use the mass action law to translate each chemical reaction into a system of differential equations. \(\mathrm{E}+\mathrm{S} \stackrel{k_{1}}{\longrightarrow} \mathrm{ES} \stackrel{k_{2}}{\longrightarrow} \mathrm{E}+\mathrm{P}\)

Solve the given initial-value problem. \(\left[\begin{array}{l}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{ll}1 & 3 \\ 0 & 2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=2\) and \(x_{2}(0)=-1\).

Drug Elimination Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time \(t\) by \(x_{1}(t)\) and in the urine at time \(t\) by \(x_{2}(t) .\) If \(x_{1}(0)=6 \mathrm{mg}\) and \(x_{2}(0)=0\), find a system of differential equations for \(x_{1}(t)\) and \(x_{2}(t)\) if it takes 20 minutes for the drug to be at one- half of its initial amount in the body.

Use the mass action law to translate each chemical reaction into a system of differential equations. \(\mathrm{A}+\mathrm{B} \stackrel{k}{\longrightarrow} \mathrm{A}+\mathrm{C}\)

Solve the given initial-value problem. \(\left[\begin{array}{c}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{rr}3 & -2 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=1\) and \(x_{2}(0)=1\).

Forest Disturbances Disturbances in forests (wind, fire, etc.) create gaps by killing trees. These gaps are eventually filled by new trees. We will model this process by a two-compartment model. We denote by \(x_{1}(t)\) the area occupied by gaps and by \(x_{2}(t)\) the area occupied by adult trees. We assume that the dynamics are given by $$ \begin{array}{l} \frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2} \\ \frac{d x_{2}}{d t}=0.2 x_{1}-0.1 x_{2} \end{array} $$ (a) Find the corresponding compartment diagram. (b) Show that \(x_{1}(t)+x_{2}(t)\) is a constant. Denote the constant by \(A\) and give its meaning. [Hint: Show that \(\left.\frac{d}{d t}\left(x_{1}+x_{2}\right)=0 .\right]\) (c) Let \(x_{1}(0)+x_{2}(0)=20\). Use your answer in (b) to explain why this equation implies that \(x_{1}(t)+x_{2}(t)=20\) for all \(t>0\). (d) Use your result in (c) to replace \(x_{2}\) in (11.51) by \(20-x_{1}\), and show that doing so reduces the system (11.51) and (11.52) to $$ \frac{d x_{1}}{d t}=2-0.3 x_{1} $$ with \(x_{1}(t)+x_{2}(t)=20\) for all \(t \geq 0\). (e) Solve the system (11.51) and (11.52), and determine what fraction of the forest is occupied by adult trees at time \(t\) when \(x_{1}(0)=2\) and \(x_{2}(0)=18\). What happens as \(t \rightarrow \infty\) ?

Show that the following system of differential equations has a conserved quantity, and find it: $$ \begin{array}{l} \frac{d x}{d t}=2 x-3 y \\ \frac{d y}{d t}=3 y-2 x \end{array} $$

Solve the given initial-value problem. \(\left[\begin{array}{l}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{rr}-1 & 0 \\ 1 & -2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=-1\) and \(x_{2}(0)=-2\).

Forest Succession Forest succession can be modeled by a three-compartment model. We assume that gaps in a forest are created by disturbances just as in Problem 21 . These gaps are initially filled by fast-growing, early colonizing plants, which are then replaced by slower growing species, a process known as succession. We denote by \(x_{1}(t)\) the total area occupied by gaps at time \(t\), by \(x_{2}(t)\) the total area occupied by fast growing species at time \(t\), and by \(x_{3}(t)\) the total area occupied by slow growing species at time \(t\). The dynamics are given by $$ \begin{array}{l} \frac{d x_{1}}{d t}=0.2 x_{2}+x_{3}-2 x_{1} \\ \frac{d x_{2}}{d t}=2 x_{1}-0.7 x_{2} \\ \frac{d x_{3}}{d t}=0.5 x_{2}-x_{3} \end{array} $$ (a) Draw the corresponding compartment diagram. (b) Show that $$ x_{1}(t)+x_{2}(t)+x_{3}(t)=A $$ where \(A\) is a constant, and give the meaning of \(A\).

Show that the following system of differential equations has a conserved quantity, and find it: $$ \begin{array}{l} \frac{d x}{d t}=-4 x+2 y \\ \frac{d y}{d t}=-y+2 x \end{array} $$

Assume that $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(10-2 x_{1}-x_{2}\right) \\ \frac{d x_{2}}{d t}=x_{2}\left(10-x_{1}-2 x_{2}\right) \end{array} $$ (a) Graph the zero isoclines. (b) Find all equilibria and classify them, by linearizing the system near each equilibrium. (c) Draw the directions of the vector field on the zero isoclines, and in the regions between the zero isoclines.

Solve the given initial-value problem. \(\left[\begin{array}{l}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{cc}4 & -7 \\ 2 & -5\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=13\) and \(x_{2}(0)=3\).

Frightened Romeo We will explore the situation where Juliet behaves as a cautious lover (see Example 5), but Romeo is so frightened he does not pick up her signals. In this case $$ \begin{array}{l} \frac{d J}{d t}=-c J+a R \\ \frac{d R}{d t}=-d R \end{array} $$ where \(a, c\), and \(d\) are all positive constants. (a) Interpret what behavior the equation for \(\frac{d R}{d t}\) models. (b) Write (11.54) as a matrix equation and find the eigenvalues of the associated matrix. (c) Based on your answer to (a), what happens to Romeo and Juliet's relationship as \(t \rightarrow \infty\) ?

Show that the following system of differential equations has a conserved quantity, and find it: $$ \begin{array}{l} \frac{d x}{d t}=-x+2 x y+z \\ \frac{d y}{d t}=-2 x y \\ \frac{d z}{d t}=x-z \end{array} $$

Assume that $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(10-x_{1}-2 x_{2}\right) \\ \frac{d x_{2}}{d t}=x_{2}\left(10-2 x_{1}-x_{2}\right) \end{array} $$ (a) Graph the zero isoclines. (b) Find all equilibria and classify them, by linearizing the system near each equilibrium. (c) Draw the directions of the vector field on the zero isoclines, and in the regions between the zero isoclines.

Solve the given initial-value problem. \(\left[\begin{array}{l}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{ll}-3 & 4 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=1\) and \(x_{2}(0)=2\).

24\. Two Hot and Cold Lovers Imagine that Romeo and Juliet both behave in the same way Romeo does in Example \(4 ;\) that is, their affections are modeled by a system of equations $$ \begin{array}{l} \frac{d J}{d t}=-b R \\ \frac{d R}{d t}=-b J \end{array} $$ where \(b\) is a positive constant. (a) By writing the system (11.55) as a matrix equation classify the equilibrium \((0,0)\); i.e., is it a stable node, spiral, saddle point, and so on? (b) Find the eigenvector directions for the equilibrium. (c) Based on your answers to (a) and (b), what is the fate of Romeo and Juliet's relationship as \(t \rightarrow \infty\) ?

Suppose that \(x(t)+y(t)\) is a conserved quantity. If $$ \frac{d x}{d t}=-3 x+2 x y $$ find the differential equation for \(y(t)\).

Let $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(2-x_{1}\right)-x_{1} x_{2} \\ \frac{d x_{2}}{d t}=x_{1} x_{2}-x_{2} \end{array} $$ (a) Graph the zero isoclines. (b) Find all equilibria and classify them, by linearizing the system near each equilibrium. (c) Draw the directions of the vector field on the zero isoclines, and in the regions between the zero isoclines.

Solve the given initial-value problem. \(\left[\begin{array}{c}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{rr}4 & 7 \\ 1 & -2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=-1\) and \(x_{2}(0)=-2\).

The Michaelis-Menten law [Equation (11.93)] states that $$ \frac{d p}{d t}=\frac{v_{m} s}{K_{m}+s} $$ where \(p=p(t)\) is the concentration of the product of the enzymatic reaction at time \(t, s=s(t)\) is the concentration of the substrate at time \(t\), and \(v_{m}\) and \(K_{m}\) are positive constants Set $$ f(s)=\frac{v_{m} s}{K_{m t}+s} $$ where \(v_{m}\) and \(K_{m}\) are positive constants. (a) Show that $$ \lim _{s \rightarrow \infty} f(s)=v_{m} $$ (b) Show that $$ f\left(K_{m}\right)=\frac{v_{m}}{2} $$ (c) Show that, for \(s \geq 0, f(s)\) is (i) nonnegative, (ii) increasing. and (iii) concave down. Sketch a graph of \(f(s) .\) Label \(v_{m}\) and \(K_{m}\) on your graph. (d) Explain why we said that the reaction rate \(d p / d t\) is limited by the availability of the substrate.

Let $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(2-x_{1}^{2}\right)-x_{1} x_{2} \\ \frac{d x_{2}}{d t}=x_{1} x_{2}-x_{2} \end{array} $$ (a) Graph the zero isoclines. (b) Find all equilibria and classify them, by linearizing the system near each equilibrium. (c) Draw the directions of the vector field on the zero isoclines, and in the regions between the zero isoclines.

Love-Struck Romeo and Juliet Romeo and Juliet are both reckless lovers; being in love intoxicates each of them and causes their love to increase, regardless of the feelings of the other. So if Romeo starts off even mildly fond of Juliet (i.e., \(R>0\) ), then his love will grow regardless of her feelings. Juliet behaves similarly. Conversely, if Romeo hates Juliet, his hatred will grow by itself. We therefore anticipate that if \(R>0\), then \(d R / d t>0\), while if \(R<0\), then \(d R / d t<0\), and similarly for \(J .\) We model Romeo and Juliet's relationship by a system of differential equations. $$ \begin{array}{l} \frac{d J}{d t}=k J \\ \frac{d R}{d t}=k R \end{array} $$ where \(k\) is a positive constant. (a) Write the system (11.57) as a matrix equation, and find the eigenvalues of the corresponding matrix. Why can we not use the methods from Section \(11.1\) to classify the equilibrium \((0,0)\) ? (b) Solve the equations (11.57) directly to find \(R(t), J(t)\) if \(R(0)=1, J(0)=-1\). What is the fate of Romeo and Juliet's relationship as \(t \rightarrow \infty\) ? (c) Try to describe generally what happens to Romeo and Juliet's relationship as \(t \rightarrow \infty\) for all possible initial conditions.

Let $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ (a) Show that $$ A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ has the repeated eigenvalues \(\lambda_{1}=\lambda_{2}=1\). (b) Show that \(\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\left[\begin{array}{l}0 \\ 1\end{array}\right]\) are eigenvectors of \(A\) and that any vector \(\left[\begin{array}{l}c_{1} \\ c_{2}\end{array}\right]\) can be written as $$ \left[\begin{array}{l} c_{1} \\ c_{2} \end{array}\right]=c_{1}\left[\begin{array}{l} 1 \\ 0 \end{array}\right]+c_{2}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$ (c) Show that $$ \mathbf{x}(t)=c_{1} e^{t}\left[\begin{array}{l} 1 \\ 0 \end{array}\right]+c_{2} e^{t}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$ is a solution of \((11.35)\) that satisfies the initial condition \(x_{1}(0)=c_{1}\) and \(x_{2}(0)=c_{2} .\)

In Problems 29-42, we consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}2 & 1 \\ 0 & 3\end{array}\right]\)

Based on each system of equations modeling Romeo and Juliet's relationship, describe in words how Romeo and Juliet are both behaving (you do not need to solve any of the systems). \(\frac{d J}{d t}=0.1 R-2 J\) \(\frac{d R}{d t}=J-0.1 R\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 4\end{array}\right]\)

Based on each system of equations modeling Romeo and Juliet's relationship, describe in words how Romeo and Juliet are both behaving (you do not need to solve any of the systems). \(\frac{d J}{d t}=J-0.2 R\) \(\frac{d R}{d t}=J-0.1 R\)

The spread of a disease through a population of 100 elephants is modeled by the following system of SIRS equations: $$ \begin{array}{l} \frac{d S}{d t}=\frac{R}{10}-\frac{1}{200} S I \\ \frac{d I}{d t}=\frac{1}{200} S I-\frac{I}{10} I \\ \frac{d R}{d t}=\frac{1}{10} I-\frac{R}{10} \end{array} $$ Find all of the equilibria for this model and classify them (e.g." as stable nodes, unstable nodes, or saddles) by analyzing the linearized system.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-2 & 2 \\ 2 & 1\end{array}\right]\)

Based on each system of equations modeling Romeo and Juliet's relationship, describe in words how Romeo and Juliet are both behaving (you do not need to solve any of the systems). \(\frac{d J}{d t}=R-2 J\) \(\frac{d R}{d t}=3 R+J\)

The spread of a disease through a herd of 50 cattle is modeled by the following system of SIRS equations: $$ \begin{array}{l} \frac{d S}{d t}=R-\frac{1}{50} S I \\ \frac{d I}{d t}=\frac{1}{50} S I-2 I \\ \frac{d R}{d t}=2 I-R \end{array} $$ Find all of the equilibria for this model and classify them (e.g., as stable nodes, unstable nodes, or saddles) by analyzing the linearized system.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}1 & 1 \\ 2 & -1\end{array}\right]\)

Solve $$ \frac{d^{2} x}{d t^{2}}=-4 x $$ with \(x(0)=0\) and \(x^{\prime}(0)=6\).

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}-4 & 2 \\ -5 & 3\end{array}\right]\)

Solve $$ \frac{d^{2} x}{d t^{2}}=-9 x $$ with \(x(0)=0\) and \(x^{\prime}(0)=12\).

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}2 & 4 \\ 2 & -2\end{array}\right]\)

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}=3 x $$ into a system of first-order differential equations.

The spread of a disease through a population of 100 individuals is represented by the following SIRS model: $$ \begin{array}{l} \frac{d S}{d t}=\frac{1}{10} R-\frac{1}{100} S I \\ \frac{d I}{d t}=\frac{1}{100} S I-\frac{1}{2} I \\ \frac{d R}{d t}=\frac{1}{2} I-\frac{1}{10} R \end{array} $$ In this problem we will sketch the directions of the solution in the SI -plane. (a) Eliminate \(R\) to rewrite the equation system as a system of differential equations in the dependent variables \(S\) and \(I\). (b) Draw the zero isoclines of your system from part (a). (c) Find all of the equilibria for this model and classify them (e.g., as stable nodes, unstable nodes, or saddles) by analyzing the linearized system. (d) Add to your plot from part (b) arrows showing the direction of the vector field on the isoclines, and in the regions between the isoclines.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}1 & 3 \\ 1 & -1\end{array}\right]\)

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}=-\frac{1}{2} x $$ into a system of first-order differential equations.

The spread of a disease through a population of 250 individuals is represented by the following SIRS model; $$ \begin{array}{l} \frac{d S}{d t}=R-\frac{1}{50} S I \\ \frac{d I}{d t}=\frac{1}{50} S I-\frac{1}{10} I \\ \frac{d R}{d t}=\frac{1}{10} I-R \end{array} $$ In this problem we will sketch the directions of the solution in the SI-plane. (a) Eliminate \(R\) to rewrite the equation system as a system of differential equations in the dependent variables \(S(t)\) and \(I(t)\). (b) Draw the zero isoclines of your system from part (a). (c) Find all of the equilibria for this model and classify them (e.g., as stable nodes, unstable nodes, or saddles) by analyzing the linearized system. (d) Add to your plot from part (b) arrows showing the direction of the vector field on the isoclines, and in the regions between the isoclines.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-1 & 3 \\ 2 & 4\end{array}\right]\)

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x $$ into a system of first-order differential equations.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-3 & -1 \\ 1 & -6\end{array}\right]\)

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}=\frac{x}{2} $$ into a system of first-order differential equations.

Some diseases are lethal; not every individual infected by the disease will recover; some will die. Assume that in one unit of time a fraction \(m\) of infected individuals will die \((m\) is called the mortality rate \() .\) (a) Explain how the SIRS model equations should be modified to incorporate deaths. In particular you should write down a new differential equation for \(\frac{d I}{d t}\). (b) Explain why it is no longer possible to eliminate \(R(t)\) from the SIRS model equations.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-3 & 1 \\ 1 & -2\end{array}\right]\)