Chapter 11

A very simple two-compartment model for gap dynamics in a forest assumes that gaps are created by disturbances (wind, fire, etc.) and that gaps revert to forest as trees grow in the gaps. 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 \(\frac{d}{d t}\left(x_{1}+x_{2}\right)=0 .\) ] (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 (11.44) by \(20-x_{1}\), and show that doing so reduces the system \((11.44)\) and \((11.45)\) 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 (11.44) and (11.45), 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 \(t \rightarrow \infty\) ?

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

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

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) Show that \((1,1)\) is an equilibrium. Use the graphical approach to determine its stability. 24\. 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) Show that \((1,1)\) is an equilibrium. Use the graphical approach to determine its stability.

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

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

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

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

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

Solve the given initial-value problem. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 2 & 6 \\ 1 & 3 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ $$ \text { with } x_{1}(0)=-3 \text { and } x_{2}(0)=1 \text { . } $$

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.

We discuss the case of repeated eigenvalues. $$ \left[\begin{array}{l} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ (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 \((11.34)\) that satisfies the initial condition \(x_{1}(0)=c_{1}\) and \(x_{2}(0)=c_{2}\).

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

We discuss the case of repeated eigenvalues. $$ \left[\begin{array}{l} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ (a) Show that $$ A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right] $$ has the repeated eigenvalues \(\lambda_{1}=\lambda_{2}=1\). (b) Show that every eigenvector of \(A\) is of the form $$ c_{1}\left[\begin{array}{l} 1 \\ 0 \end{array}\right] $$ where \(c_{1}\) is a real number different from \(0 .\) (c) Show that $$ \mathbf{x}_{1}(t)=e^{t}\left[\begin{array}{l} 1 \\ 0 \end{array}\right] $$ is a solution of \((11.35)\).

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}=3 x $$ 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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} 2 & -1 \\ 0 & 3 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -1 & 0 \\ 0 & -4 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -2 & 2 \\ 2 & 1 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -5 & -2 \\ 6 & 3 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{ll} -4 & 2 \\ -5 & 3 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point.$$ A=\left[\begin{array}{rr} -2 & 4 \\ 2 & -5 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} 6 & -4 \\ -3 & 5 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -1 & 3 \\ 2 & -5 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point.3 $$ A=\left[\begin{array}{rr} -3 & -1 \\ 1 & -6 \end{array}\right] $$

Consider communities composed of two species. The abundance of species 1 at time \(t\) is given by \(N_{1}(t)\), the abundance of species 2 at time \(t\) by \(N_{2}(t) .\) Their dynamics are described by $$ \begin{array}{l} \frac{d N_{1}}{d t}=f_{1}\left(N_{1}, N_{2}\right) \\ \frac{d N_{2}}{d t}=f_{2}\left(N_{1}, N_{2}\right) \end{array} $$ Assume that when both species are at low abundances their abundances increase and that \(f_{1}\) and \(f_{2}\) change sign when crossing their zero isoclines. In each problem, determine the sign structure of the community matrix at the nontrivial equilibrium (indicated by a dot) on the basis of the graph of the zero isoclines. Determine the stability of the equilibria if possible.

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -3 & 1 \\ 1 & -2 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} 0 & -2 \\ -1 & 3 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{ll} 0 & 2 \\ 3 & 7 \end{array}\right] $$

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} -2 & -3 \\ 1 & 3 \end{array}\right] $$

Assume that the diagonal elements \(a_{i i}\) of the community matrix of a species assemblage in equilibrium are negative. Explain why this assumption implies that species \(i\) exhibits selfregulation.

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 equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{ll} 4 & -1 \\ 5 & -1 \end{array}\right] $$

Consider a community composed of two species. Assume that both species inhibit themselves. Explain why mutualistic and competitive interactions lead to qualitatively similar predictions about the stability of the corresponding equilibria. That is, show that if \(A=\left[a_{i j}\right]\) is the community matrix at equilibrium for the case of mutualism, and if \(B=\left[b_{i j}\right]\) is the community matrix at equilibrium for the case of competition, then the following holds: If \(\left|a_{i j}\right|=\left|b_{i j}\right|\) for \(1 \leq i, j \leq 2\), then either both equilibria are Iocally stable or both are unstable

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} 2 & -1 \\ 3 & 0 \end{array}\right] $$

The classical Lotka-Volterra model of predation is given by $$ \begin{array}{l} \frac{d N}{d t}=a N-b N P \\ \frac{d P}{d t}=c N P-d P \end{array} $$ where \(N=N(t)\) is the prey density at time \(t\) and \(P=P(t)\) is the predator density at time \(t .\) The constants \(a, b, c\), and \(d\) are all positive. (a) Find the nontrivial equilibrium \((\hat{N}, \hat{P})\) with \(\hat{N}>0\) and \(\hat{P}>0\). (b) Find the community matrix corresponding to the nontrivial equilibrium. (c) Explain each entry of the community matrix found in (b) in terms of how individuals in this community affect each other.

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} -1 & -5 \\ 4 & -3 \end{array}\right] $$

The modified Lotka-Volterra model of predation is given by $$ \begin{array}{l} \frac{d N}{d t}=a N\left(1-\frac{N}{K}\right)-b N P \\ \frac{d P}{d t}=c N P-d P \end{array} $$ where \(N=N(t)\) is the prey density at time \(t\) and \(P=P(t)\) is the predator density at time \(t\). The constants \(a, b, c, d\), and \(K\) are positive. Assume that \(d / c0\) and \(\hat{P}>0\). (b) Find the community matrix corresponding to the nontrivial equilibrium. (c) Explain each entry of the community matrix found in (b) in terms of how individuals in this community affect each other.

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} -2 & 4 \\ -3 & -2 \end{array}\right] $$

Use a graphing calculator to study the following example of the Fitzhugh- Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-0.3)(V-1)-w \\ \frac{d w}{d t}=0.01(V-0.4 w) \end{array} $$ Sketch the graph of the solution curve in the \(V-w\) plane when (i) \((V(0), w(0))=(0.4,0)\) and (ii) \((V(0), w(0))=(0.2,0)\).

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{ll} 1 & 3 \\ -2 & 1 \end{array}\right] $$

Use a graphing calculator to study the following example of the Fitzhugh- Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-0.6)(V-1)-w \\ \frac{d w}{d t}=0.03(V-0.6 w) \end{array} $$ Sketch the graph of the solution curve in the \(V-w\) plane when (i) \((V(0), w(0))=(0.8,0)\) and (ii) \((V(0), w(0))=(0.4,0)\).

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{ll} 2 & -3 \\ 2 & -1 \end{array}\right] $$

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} 4 & 5 \\ -3 & -3 \end{array}\right] $$

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

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{ll} -1 & 1 \\ -3 & 1 \end{array}\right] $$

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{ll} -1 & 1 \\ -3 & 1 \end{array}\right] $$

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

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} 3 & -2 \\ 1 & 3 \end{array}\right] $$

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

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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right] $$

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