91Ó°ÊÓ

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{\prime \prime}+\left(x^{\prime}\right)^{2}+2 x=0\)

Show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=-x+y^{2} \\ &y^{\prime}=x-y \end{aligned} $$

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x^{\prime}\left(1-x^{3}\right)-x^{2}=0 $$

Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$\frac{d x}{d t}=k x(n+1-x)$$

The general solution of the linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}\) is given. (a) In each case discuss the nature of the solution in a neighborhood of \((0,0)\). (b) With the aid of a graphing utility plot the solution that satisfies \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right), \quad \mathbf{X}(t)=e^{t}\left[c_{1}\left(\begin{array}{c} -\sin t \\ \cos t \end{array}\right)+c_{2}\left(\begin{array}{c} \cos t \\ \sin t \end{array}\right)\right] $$

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{\prime \prime}+x^{\prime}\left(1-x^{3}\right)-x^{2}=0\)

In Problems \(3-10\), without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=k x(n+1-x) $$

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=-x+y^{2} \\ &y^{\prime}=x-y \end{aligned} $$

In Problems \(3-10\), without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=-k x \ln \frac{x}{k}, x>0 $$

The general solution of the linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}\) is given. (a) In each case discuss the nature of the solution in a neighborhood of \((0,0)\). (b) With the aid of a graphing utility plot the solution that satisfies \(\mathbf{X}(0)=(1,1)\) $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} -1 & -4 \\ 1 & -1 \end{array}\right) \\ &\mathbf{X}(t)=e^{-t}\left[c_{1}\left(\begin{array}{c} 2 \cos 2 t \\ \sin 2 t \end{array}\right)+c_{2}\left(\begin{array}{c} -2 \sin 2 t \\ \cos 2 t \end{array}\right)\right] \end{aligned} $$