Chapter 10

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{1-6 \text { Each of the nonlinear systems has an equilibrium at }} \\ {\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) \text { . Find the linearization near this point. }}\end{array}$$ $$\begin{array}{l}{\frac{d x_{1}}{d t}=4 x_{1}-2 x_{1} x_{2}} \\ {\frac{d x_{2}}{d t}=-2 x_{2}+8 x_{1} x_{2}}\end{array}$$

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{cc}{3} & {-2} \\ {2} & {-2}\end{array}\right]} \\ {x_{1}(t)=\frac{1}{3}\left(4 e^{2 t}-e^{-t}\right),} & {x_{2}(t)=\frac{2}{3}\left(e^{2 t}-e^{-t}\right)}\end{array}\)

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d x / d t=x-y, \quad d y / d t=-3 t y+x\)

2\. Hemodialysis is a process by which a machine is used to filter urea and other waste products from a patient's blood if the kidneys fail. The amount of urea within a patient during dialysis is sometimes modeled by supposing there are two compartments within the patient: the blood, which is directly filtered by the dialysis machine, and another com- partment that cannot be directly filtered but that is con- nected to the blood. A system of two differential equations describing this is $$\frac{d c}{d t}=-\frac{K}{V} c+a p-b c \quad \frac{d p}{d t}=-a p+b c$$ where \(c\) and \(p\) are the urea concentrations in the blood and the inaccessible pool (in \(\mathrm{mg} / \mathrm{mL} )\) and all constants are positive (see also Exercise 14 in the Review Section of this chapter). Suppose that \(K / V=1, a=b=\frac{1}{2},\) and the initial urea concentration is \(c(0)=c_{0}\) and \(p(0)=c_{0} \mathrm{mg} / \mathrm{mL}\) $$\begin{array}{l}{\text { (a) Classify the equilibrium of this system. }} \\\ {\text { (b) Solve this initial-value problem. }}\end{array}$$

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{\frac{d x_{1}}{d t}=4 x_{1}\left(1-5 x_{1}\right)-2 x_{1} x_{2}} \\ {\frac{d x_{2}}{d t}=-2 x_{2}+8 x_{1} x_{2}}\end{array}$$

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{rr}{-2} & {1} \\ {1} & {-2}\end{array}\right]} \\ {x_{1}(t)=\frac{1}{2}\left(e^{-3 t}+e^{-1}\right),} & {x_{2}(t)=\frac{1}{2}\left(-e^{-3 t}+e^{-t}\right)}\end{array}\)

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d y / d x=2 y, \quad d z / d x=x-z+3\)

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{\frac{d x_{1}}{d t}=\sin x_{1}+x_{1} x_{2}+3 x_{2}^{2}} \\\ {\frac{d x_{2}}{d t}=\cos x_{2}-1+x_{1}\left(x_{1}-1\right)+7 x_{2}}\end{array}$$

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{rr}{-1} & {-4} \\ {1} & {-1}\end{array}\right]} \\ {x_{1}(t)=e^{-t} \cos 2 t,} & {x_{2}(t)=\frac{1}{2} e^{-t} \sin 2 t}\end{array}\)

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d y / d t=3 y z-2 z, \quad d z / d t=2 z+5 y\)

4\. Soil contamination A crop is planted in soil that is contaminated with a pollutant. The pollutant gradually leaches out of the soil but is also absorbed by the growing crop. A simple model of this process is $$\frac{d s}{d t}=-\alpha s-\beta s \quad \frac{d c}{d t}=\beta s$$ where \(s\) and \(c\) are the amounts of pollutant in the soil and crop (in mg), respectively, and \(\alpha\) and \(\beta\) are positive constants. $$\begin{array}{l}{\text { (a) Suppose that } s(0)=s_{0} \text { and } c(0)=0 . \text { What is the solu- }} \\ {\text { tion to the initial-value problem? }} \\ {\text { (b) In the long term (that is, as } t \rightarrow \infty ), \text { what is the amount }} \\ {\text { of pollutant in the crop? }}\end{array}$$

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{\frac{d x_{1}}{d t}=5\left(1+\cos x_{1}\right)+a x_{1}-b x_{2}-10} \\ {\frac{d x_{2}}{d t}=3 x_{2}+b x_{1} x_{2}}\end{array}$$

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{rr}{1} & {2} \\ {-4} & {1}\end{array}\right]} \\ {x_{1}(t)=e^{t} \cos (2 \sqrt{2} t),} & {x_{2}(t)=-\sqrt{2} e^{t} \sin (2 \sqrt{2} t)}\end{array}\)

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d y / d x=3 y-2, \quad d z / d x=7 z+y\)

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{\frac{d x_{1}}{d t}=1+x_{1}^{3}-\frac{1+x_{1}}{1+x_{2}}} \\\ {\frac{d x_{2}}{d t}=2 x_{2}+x_{1}^{2}}\end{array}$$

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d x / d z=3 x-2 y, \quad d y / d z=2 z+3 y\)

6\. Radioimmunotherapy is a cancer treatment in which radioactive atoms are attached to tumor-specific antibody molecules and then injected into the blood. The antibody molecules then attach only to tumor cells, where they then deliver the cell-killing radioactivity. The following model for this process was introduced in Example 10.1.1: $$\frac{d x_{1}}{d t}=-a x_{1}-b x_{1} \quad \frac{d x_{2}}{d t}=b x_{1}-c x_{2}$$ $$\begin{array}{l}{\text { where } x_{1} \text { denotes the amount of antibody in the blood and } x_{2}} \\ {\text { the amount of antibody taken up by the tumor (both in } \mu g ) .} \\ {\text { All constants are positive. }} \\ {\text { (a) Find the general solution. }} \\ {\text { (b) Suppose that } x_{1}(0)=x_{0} \text { and } x_{2}(0)=0 . \text { What is the }} \\\ {\text { solution to this initial-value problem? }}\end{array}$$

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{\frac{d x_{1}}{d t}=x_{2}-\frac{2+a x_{1}}{2+b x_{2}}+\cos x_{2}} \\ {\frac{d x_{2}}{d t}=\frac{2 x_{1}}{1+x_{2}}-a x_{1}}\end{array}$$

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. \(d x / d t=x y-y, \quad d y / d t=4 t x-x y\)

7\. Cancer progression The development of many cancers, such as colorectal cancer, proceed through a series of pre- cancerous stages. Suppose there are \(n-1\) precancerous stages before developing into cancer at stage \(n .\) A simple system of differential equations modeling this is $$\begin{aligned} x_{0}^{\prime} &=-u_{0} x_{0} \\ x_{i}^{\prime} &=u_{i-1} x_{i-1}-u_{i} x_{i} \\ x_{n}^{\prime} &=u_{n-1} x_{n-1} \end{aligned}$$ where \(x_{i}\) is the fraction of the population in state \(i,\) the \(u_{i}\) 's are positive constants, and \(i=1, \ldots, n-1 .\) $$\begin{array}{l}{\text { (a) Suppose } n=2 . \text { What is the system of differential equa }} \\ {\text { tions for the three stages? }} \\ {\text { (b) Note that the variable } x_{2} \text { does not appear in the equa- }} \\\ {\text { tions for the rate of change of } x_{0} \text { or } x_{1} . \text { Consequently, }} \\ {\text { we can solve the two-dimensional system for } x_{0} \text { and }} \\ {x_{1} \text { separately. Do so, assuming that } x_{0}(0)=k \text { and }} \\ {x_{1}(0)=0 .}\end{array}$$ $$\begin{array}{l}{\text { (c) Use your solution for } x_{1}(t) \text { obtained in part (b) to write }} \\ {\text { a differential equation for } x_{2}(t) \text { . }} \\ {\text { (d) Solve the differential equation from part (c), assuming }} \\ {x_{2}(0)=0}\end{array}$$

Find all equilibria. Then find the linearization near each equilibrium. $$\begin{array}{l}{\frac{d x_{1}}{d t}=-5 x_{1}+x_{1} x_{2}} \\ {\frac{d x_{2}}{d t}=x_{2}-5 x_{1} x_{2}}\end{array}$$

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the initial-value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(\begin{array}{l}{A=\left[ \begin{array}{rr}{1} & {-1} \\ {1} & {1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{2} \\\ {1}\end{array}\right]} \\ {x_{1}(t)=e^{t}(2 \cos t-\sin t),} & {x_{2}(t)=e^{t}(\cos t+2 \sin t)}\end{array}\)

Write each system of linear differential equations in matrix notation. \(d x / d t=5 x-3 y, \quad d y / d t=2 y-x\)

8\. Metapopulations Consider a simple metapopulation in which subpopulation \(\mathrm{A}\) grows at a per capita rate of \(r_{\mathrm{A}}=1\) and subpopulation \(\mathrm{B}\) declines at a per capita rate of \(r_{\mathrm{B}}=-1 .\) Suppose the per capita rate of movement between subpopulation patches is \(m\) in both directions. This gives $$\begin{aligned} \frac{d x_{\mathrm{A}}}{d t} &=(1-m) x_{\mathrm{A}}+m x_{\mathrm{B}} \\ \frac{d x_{\mathrm{B}}}{d t} &=-(1+m) x_{\mathrm{B}}+m x_{\mathrm{A}} \end{aligned}$$ where \(x_{\text { A }}\) and \(x_{\mathrm{B}}\) are the numbers of individuals in patches \(\mathrm{A}\)and \(\mathrm{B},\) respectively. $$\begin{array}{l}{\text { (a) Classify the equilibrium at the origin. }} \\\ {\text { (b) Find the general solution. }} \\ {\text { (c) What is the solution to the initial-value problem if }} \\ {x_{\mathrm{A}}(0)=1 \text { and } x_{\mathrm{B}}(0)=0 ?}\end{array}$$

Find all equilibria. Then find the linearization near each equilibrium. $$\begin{array}{l}{\frac{d x_{1}}{d t}=x_{2}-5 x_{1} x_{2}} \\ {\frac{d x_{2}}{d t}=2 x_{1}-6 x_{1} x_{2}}\end{array}$$

Write each system of linear differential equations in matrix notation. \(d x / d t=x-2, \quad d y / d t=2 y+3 x-1\)

9\. Suppose a glass of cold water is sitting in a warm room and you place a coin at room temperature \(R\) into the glass. The coin gradually cools down while, at the same time, the glass of water warms up. Newton's law of cooling suggests the following system of differential equations to describe the process $$\frac{d w}{d t}=-k_{m}(w-R) \quad \frac{d p}{d t}=-k_{p}(p-w)$$ where \(w\) and \(p\) are the temperatures of the water and coil (in "C), respectively, and the \(k\) 's are positive constants. $$\begin{array}{l}{\text { (a) Explain the form of the system of differential equations }} \\ {\text { and the assumptions that underlie them. }} \\\ {\text { (b) Use a change of variables to obtain a homogeneous }} \\ {\text { system. }} \\ {\text { (c) What is the general solution to the system you found in }} \\ {\text { part (b)? }} \\ {\text { (d) What is the solution to the original initial-value problem }} \\ {\text { if } w(0)=w_{0} \text { and } p(0)=p_{0} ?}\end{array}$$

Find all equilibria. Then find the linearization near each equilibrium. $$\begin{array}{l}{\frac{d x_{1}}{d t}=x_{1}-6 x_{2}^{2}+x_{1} x_{2}} \\\ {\frac{d x_{2}}{d t}=8 x_{1}+4 x_{1} x_{2}}\end{array}$$

Write each system of linear differential equations in matrix notation. \(d x / d t=3 t v-7, \quad d v / d t=2 x-3 y\)

Find all equilibria. Then find the linearization near each equilibrium. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}-2 x_{1}^{2}-6 x_{1} x_{2} \\ \frac{d x_{2}}{d t}=2 x_{2}-8 x_{2}^{2}-2 x_{1} x_{2} \end{array} $$

Show that if the eigenvalues of a \(2 \times 2\) matrix are real and distinct, then the matrix \(P\) whose columns are the corresponding eigenvectors is nonsingular.

Write each system of linear differential equations in matrix notation. \(d x / d t=5 y, \quad d y / d t=2 x-y\)

Find all equilibria. Then find the linearization near each equilibrium. $$\begin{array}{l}{\frac{d x_{1}}{d t}=e^{-x_{1}}\left(x_{1}-x_{2}\right)} \\\ {\frac{d x_{2}}{d t}=x_{1}-x_{2}^{2}+2 x_{1} x_{2}}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=-1, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-1} \\ {1}\end{array}\right]\)

Write each system of linear differential equations in matrix notation. \(d x / d t=2 x-5, \quad d y / d t=3 x+7 y\)

12\. Systemic lupus erythematosus is an autoimmune disease in which some immune molecules, called antibodies, target DNA instead of pathogens. This can be treated by injecting drugs that absorb the offending antibodies. The antibodies are found in both the bloodstream and in organs, and this can be modeled using a two-compartment model: A system of differential equations describing the amount of antibody in each compartment is $$\begin{aligned} \frac{d x_{1}}{d t} &=G+k_{21} x_{2}-k_{12} x_{1}-k x_{1} \\\ \frac{d x_{2}}{d t} &=k_{12} x_{1}-k_{21} x_{2} \end{aligned}$$ where \(G\) is the rate of generation of antibodies, \(k\) is the rate at which the drug treatment removes antibody from the bloodstream, and \(k_{i j}\) is the rate of flow of antibody from compartment \(i\) to \(j .\) The variables \(x_{1}\) and \(x_{2}\) are the amounts of antibody in the bloodstream and organs, respectively, measured in \(\mu \mathrm{g}\) . (See also Exercise 16 in the Review Sec- tion of this chapter.) $$\begin{array}{l}{\text { (a) Use a change of variables to obtain a homogene- }} \\ {\text { ous system of differential equations describing the }} \\\ {\text { situation. }} \\ {\text { (b) What is the general solution to the differential equa- }} \\ {\text { tions in part (a)? }} \\ {\text { (c) What is the general solution obtained in part (b) in }} \\ {\text { terms of the original variables } x_{1} \text { and } x_{2} ?}\end{array}$$

Find all equilibria. Then find the linearization near each equilibrium. $$\begin{array}{l}{\frac{d x_{1}}{d t}=\ln x_{1}-x_{2}} \\ {\frac{d x_{2}}{d t}=x_{1}\left(1-x_{1}-x_{2}\right)}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=2, \quad \lambda_{2}=4 ; \quad \quad v_{1}=\left[ \begin{array}{l}{1} \\ {2}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{l}{0} \\ {1}\end{array}\right]\)

Write each system of linear differential equations in matrix notation. \(d x / d t=2 x-y \sin t, \quad d y / d t=y-x\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{\left(x_{1}-2\right) x_{2}+x_{1} x_{2}} & {x_{1}\left(x_{1}-2\right)} \\ {0} & {-1+2 x_{2}}\end{array}\right]} \\\ {\text { (i) } \hat{x}_{1}=0, \hat{x}_{2}=2} \\ {\text { (ii) } \hat{x}_{1}=2, \hat{x}_{2}=-1}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=2, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{3} \\ {1}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right]\)

Write each system of linear differential equations in matrix notation. \(d x / d t=x+4 y-3 t, \quad d y / d t=y-x\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{-1+2 x_{1}} & {0} \\ {0} & {-\frac{1}{3}+2 x_{2}}\end{array}\right]} \\ {\text { (i) } \hat{x}_{1}=-1, \hat{x}_{2}=0} \\ {\text { (ii) } \hat{x}_{1}=2, \hat{x}_{2}=\frac{1}{3}}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=-3, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{l}{0} \\ {1}\end{array}\right]\)

Write each system of linear differential equations in matrix notation. \(d x / d t=y-2 x \sqrt{t}+7, \quad d y / d t=3 x+2\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{1-\cos x_{2}} & {\left(x_{1}-1\right) \sin x_{2}} \\ {\cos x_{1}} & {-\sin 1}\end{array}\right]} \\ {\text { (i) } \hat{x}_{1}=0, \hat{x}_{2}=0} \\\ {\text { (ii) } \hat{x}_{1}=1, \hat{x}_{2}=1}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=5, \quad \lambda_{2}=1 ; \quad \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{2} \\ {2}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-2} \\ {1}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{-3} & {1} \\ {2} & {-2}\end{array}\right]\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{2 x_{1}} & {-\sin x_{2}} \\\ {\cos x_{1}} & {0}\end{array}\right]} \\ {\text { (i) } \hat{x}_{1}=1, \hat{x}_{2}=-\pi} \\ {\text { (ii) } \hat{x}_{1}=1, \hat{x}_{2}=\pi}\end{array}$$

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=1, \quad \lambda_{2}=-1 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{3} \\ {2}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-4} \\ {1}\end{array}\right]\)