Chapter 11

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

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 \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{ll} -2 & -2 \\ -2 & -5 \end{array}\right), \quad \mathbf{X}(t)=c_{1}\left(\begin{array}{r} 2 \\ -1 \end{array}\right) e^{-t}+c_{2}\left(\begin{array}{l} 1 \\ 2 \end{array}\right) e^{-6 t} $$

Show that \((0,0)\) is an asymptotically stable critical point of the nonlinear autonomous system $$ \begin{aligned} &x^{\prime}=\alpha x-\beta y+y^{2} \\ &y^{\prime}=\beta x+\alpha y-x y \end{aligned} $$ when \(\alpha<0\) and an unstable critical point when \(\alpha>0\). [Hint: Switch to polar coordinates.]

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+9 \sin x=0 $$

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

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{\prime \prime}+9 \sin x=0\)

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=2 x-x y \\ &y^{\prime}=-1-x^{2}+2 x-y^{2} \end{aligned} $$

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 \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{rr} -1 & -2 \\ 3 & 4 \end{array}\right), \quad \mathbf{X}(t)=c_{1}\left(\begin{array}{r} 1 \\ -1 \end{array}\right) e^{t}+c_{2}\left(\begin{array}{r} -4 \\ 6 \end{array}\right) e^{2 t} $$

When expressed in polar coordinates, a plane autonomous system takes the form $$ \begin{aligned} &\frac{d r}{d t}=\alpha r(5-r) \\ &\frac{d \theta}{d t}=-1 \end{aligned} $$ Show that \((0,0)\) is an asymptotically stable critical point if and only if \(\alpha<0\).

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

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

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

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 \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right), \quad \mathbf{X}(t)=e^{t}\left[c_{1}\left(\begin{array}{c} -\sin t \\ \cos t \end{array}\right)+c_{2}\left(\begin{array}{c} \cos t \\ \sin t \end{array}\right)\right] $$

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

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

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. $$\frac{d x}{d t}=k x(n+1-x)$$

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

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 \(\mathbf{X}(0)=(1,1)\) $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} -1 & -4 \\ 1 & -1 \end{array}\right) \\ &\mathbf{X}(t)=e^{-t}\left[c_{1}\left(\begin{array}{c} 2 \cos 2 t \\ \sin 2 t \end{array}\right)+c_{2}\left(\begin{array}{c} -2 \sin 2 t \\ \cos 2 t \end{array}\right)\right] \end{aligned} $$

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. $$ \frac{d x}{d t}=-k x \ln \frac{x}{k}, x>0 $$

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+4 \frac{x}{1+x^{2}}+2 x^{\prime}=0 $$

Show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=x y^{2}-x^{2} y \\ &y^{\prime}=x^{2} y-1 \end{aligned} $$

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. $$\frac{d x}{d t}=-k x \ln \frac{x}{k}, x>0$$

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{\prime \prime}+4 \frac{x}{1+x^{2}}+2 x^{\prime}=0\)

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=-\mu x-y \\ &y^{\prime}=x+y^{3} \\ &\text { for } \mu<0 \end{aligned} $$

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 \(\mathbf{X}(0)=(1,1)\) $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} -6 & 5 \\ -5 & 4 \end{array}\right) \\ &\mathbf{X}(t)=c_{1}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) e^{-t}+c_{2}\left[\left(\begin{array}{l} 1 \\ 1 \end{array}\right) t e^{-t}+\left(\begin{array}{l} 0 \\ \frac{1}{5} \end{array}\right) e^{-t}\right] \end{aligned} $$

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. $$ \frac{d T}{d t}=k\left(T-T_{0}\right) $$

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x=\epsilon x^{3} \text { for } \epsilon>0 $$

Without referring back to the text. Fill in the blank or answer true/false. If the critical point \((0,0)\) of the linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\) is a saddle point and \(\mathbf{X}=\mathbf{X}(t)\) is a solution, then \(\lim _{\mu_{\infty} \infty} \mathbf{X}(t)\) does not exist. _____.

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. $$\frac{d T}{d t}=k\left(T-T_{0}\right)$$

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{n}+x=\epsilon x^{3}\) for \(\epsilon>0\)

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 \(\mathbf{X}(0)=(1,1)\) $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} 2 & 4 \\ -1 & 6 \end{array}\right) \\ &\mathbf{X}(t)=c_{1}\left(\begin{array}{l} 2 \\ 1 \end{array}\right) e^{4 t}+c_{2}\left[\left(\begin{array}{l} 2 \\ 1 \end{array}\right) t e^{4 t}+\left(\begin{array}{l} 1 \\ 1 \end{array}\right) e^{4 t}\right] \end{aligned} $$

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. $$ m \frac{d v}{d t}=m g-k v $$

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x-\epsilon x|x|=0 \text { for } \epsilon>0 $$

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

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. $$m \frac{d v}{d t}=m g-k v$$

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

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ x^{\prime \prime}-2 x+\left(x^{\prime}\right)^{4}=0 $$

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 \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{ll} 2 & -1 \\ 3 & -2 \end{array}\right), \quad \mathbf{X}(t)=c_{1}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) e^{t}+c_{2}\left(\begin{array}{l} 1 \\ 3 \end{array}\right) e^{-t} $$

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. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x), \alpha>\beta $$

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x+x y \\ &y^{\prime}=-y-x y \end{aligned} $$

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. $$\frac{d x}{d t}=k(\alpha-x)(\beta-x), \alpha>\beta$$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x+x y \\ &y^{\prime}=-y-x y \end{aligned} $$

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. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x)(\gamma-x), \alpha>\beta>\gamma $$

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=y^{2}-x \\ &y^{\prime}=x^{2}-y \end{aligned} $$

Without referring back to the text. Fill in the blank or answer true/false. All solutions to the pendulum equation \(\frac{d^{2} \theta}{d t^{2}}+\frac{g}{l} \sin \theta=0\) are periodic. ______.

Show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ x^{\prime \prime}+x=\left[\frac{1}{2}+3\left(x^{\prime}\right)^{2}\right] x^{\prime}-x^{2} $$

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. $$\frac{d x}{d t}=k(\alpha-x)(\beta-x)(\gamma-x), \alpha>\beta>\gamma$$

The general solution of the linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\) 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 \(\mathbf{X}(0)=(1,1)\). $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} -1 & 5 \\ -1 & 1 \end{array}\right) \\ &\mathbf{X}(t)=c_{1}\left(\begin{array}{c} 5 \cos 2 t \\ \cos 2 t-2 \sin 2 t \end{array}\right)+c_{2}\left(\begin{array}{c} 5 \sin 2 t \\ 2 \cos 2 t+\sin 2 t \end{array}\right) \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=y^{2}-x \\ &y^{\prime}=x^{2}-y \end{aligned} $$