91Ó°ÊÓ

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=y(3 x-2), \quad y^{\prime}=2 x+9 y^{2}$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rr} 1 & 1 \\ -1 & 3 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=x+3 y^{2}, \quad y^{\prime}=y(x-2)$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 1 & 3 \\ 1 & -1 \end{array}\right]$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\mathbf{x}_{1}(t)=\left[\begin{array}{c} t \sin t \\ \cos t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} -t \cos t \\\ \sin t \end{array}\right]$$, $$A=\left[\begin{array}{rr} 1 / t & t \\ -1 / t & 0 \end{array}\right]$$.

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} -2 & 3 \\ -3 & -2 \end{array}\right]$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right]$$

Consider the predator-prey model $$\frac{d x}{d t}=x(2-y), \frac{d y}{d t}=y(x-2)$$ Sketch the phase plane for \(0 \leq x \leq 10,0 \leq y \leq 10\) Compare the behavior of the two specific cases corresponding to the initial conditions \(x(0)=1, y(0)=\) \(0.1,\) and \(x(0)=1, y(0)=1\)

Consider the predator-prey model $$\frac{d x}{d t}=x(3-x-y), \frac{d y}{d t}=y(x-1)$$ Sketch the phase plane for \(0 \leq x \leq 4,0 \leq y \leq 4\) What happens to the populations of both species as \(t \rightarrow+\infty ?\)

Describe the behavior of the solutions to \(\mathbf{x}^{\prime}=A \mathbf{x},\) if \(A=\left[\begin{array}{rl}a & b \\ -b & a\end{array}\right],\) where \(a<0\) and \(b>0\).