Deck 10: Plane Autonomous Systems

Question 1

The geometric configuration of the solutions of $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - y$ in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Answer 1

saddle point

Question 2

Assume a bead of mass $m$ slides along the curve $y = f ( x )$ . Also assume that there is a damping force acting in the direction opposite to the velocity and proportional to the velocity, with proportionality constant $\beta$ . The differential equation that describes the horizontal position of the bead is

A) $m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }$
B) $m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }$
C) $m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }$
D) $m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }$
E) $m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }$

Answer 2

$m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }$

Question 3

The values of $C$ that make the system $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = - c x - y$ stable are

A) $c > - 3 / 2$
B) $c < - 3 / 2$
C) It is locally stable for all values of $C$ .
D) It is unstable for all values of $C$ .
E) $c > 0$

Answer 3

$c > - 3 / 2$

Question 4

The constant solution of $\frac { d x } { d t } = 2 x ^ { 2 } + y ^ { 2 } - 1 , \frac { d y } { d t } = x - 2 y$ are

A) $x = 2 / 3 , y = 1 / 3$
B) $x = 2 / 3 , y = 1 / 3$ and $x = - 2 / 3 , y = - 1 / 3$
C) $x = - 2 / 3 , y = - 1 / 3$
D) $x = 2 / \sqrt { 7 } , y = 1 / \sqrt { 7 }$
E) $x = 2 / \sqrt { 7 } , y = 1 / \sqrt { 7 }$ and $x = - 2 / \sqrt { 7 } , y = - 1 / \sqrt { 7 }$

Question 5

Which of the following systems are linear? Select all that apply.

A) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 t - 3 \sin t$
B) $\frac { d x } { d t } = 3.5 x + 2 y , \frac { d y } { d t } = 2 x - 5 y$
C) $\frac { d x } { d t } = x + 1 / y , \frac { d y } { d t } = 2 x e ^ { t } - 3 y$
D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \cos x - 3 y$
E) $\frac { d x } { d t } = t ^ { 2 } + 1 , \frac { d y } { d t } = 15 x - 4 y$

Question 6

The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 y$ is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 7

The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 y$ is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 8

The solution of the system $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 y$ is

A) $x = c _ { 1 } e ^ { - 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) + 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$
B) $x = c _ { 1 } e ^ { 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { 2 t } \sin ( 2 t )$
C) $x = c _ { 1 } e ^ { - 2 t } ( 4 \cos ( 2 t ) + 2 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( - 2 \cos ( 2 t ) + 4 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$
D) $x = c _ { 1 } e ^ { - 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$
E) $x = c _ { 1 } e ^ { 2 t } ( 4 \cos ( 2 t ) + 2 \sin ( 2 t ) ) + c _ { 2 } e ^ { 2 t } ( - 2 \cos ( 2 t ) + 4 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { 2 t } \sin ( 2 t )$

Question 9

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 8 y$ at the critical point $( 2,0 )$

A) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 3 y ^ { 2 } \\8 & 0\end{array} \right]$
B) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 3 y ^ { 2 } \\0 & 8\end{array} \right]$
C) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & 3 y ^ { 2 } \\0 & 8\end{array} \right]$
D) $A = \left[ \begin{array} { c c } 12 & 0 \\0 & 8\end{array} \right]$
E) $A = \left[ \begin{array} { c c } 12 & 0 \\8 & 0\end{array} \right]$

Question 10

The solution of the system $\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 y$ is

A) $x = c _ { 1 } e ^ { - 6 t } + c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { - 6 t } - 3 c _ { 2 } e ^ { - 2 t }$
B) $x = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { 2 t } , y = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { 2 t }$
C) $x = c _ { 1 } e ^ { - 6 t } - 3 c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { - 6 t } + c _ { 2 } e ^ { - 2 t }$
D) $x = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { - 2 t }$
E) $x = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { - 2 t }$

Question 11

The initial value problem $x ^ { \prime \prime } - x ^ { \prime } + t x = 0 , x ( 0 ) = 1 , x ^ { \prime } ( 0 ) = 2$ can be rewritten as the system

A) $x ^ { \prime } = x , u ^ { \prime } = u - t x , x ( 0 ) = 1 , u ( 0 ) = 2$
B) $x ^ { \prime } = x , u ^ { \prime } = x - t u , x ( 0 ) = 1 , u ( 0 ) = 2$
C) $x ^ { \prime } = u , u ^ { \prime } = u + t x , x ( 0 ) = 1 , u ( 0 ) = 2$
D) $x ^ { \prime } = u , u ^ { \prime } = x + t u , x ( 0 ) = 1 , u ( 0 ) = 2$
E) $x ^ { \prime } = u , u ^ { \prime } = u - t x , x ( 0 ) = 1 , u ( 0 ) = 2$

Question 12

The geometric configuration of the solutions of $\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 y$ in the phase plane is

A) stable spiral point
B) unstable spiral point
C) stable node
D) unstable node
E) saddle point

Question 13

The critical points of the system $\frac { d x } { d t } = 2 x + y - 3 , \frac { d y } { d t } = 2 x - 3 y - 7$ are

A) $y = - 1$
B) $x = 2$
C) $( 2 , - 1 )$
D) $( - 1,2 )$
E) $( - 3 , - 7 )$

Question 14

Consider the differential equation $x ^ { \prime } = \sec x$ . The point $x = 0$ is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 15

The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - y$ is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 16

Which of the following systems are autonomous? Select all that apply.

A) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 t - 3 \sin t$
B) $$\frac { d x } { d t } = 3.5 x + 2 y , \frac { d y } { d t } = 2 x - 5 y$$
C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x e ^ { t } - 3 y$
D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \cos x - 3 y$
E) $\frac { d x } { d t } = t ^ { 2 } + 1 , \frac { d y } { d t } = 15 x - 4 y$

Question 17

The critical points of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } - 8 , y ^ { \prime } = 8 y$ are

A) $( 2,0 ) , 0,2 )$
B) $$( - 2,0 ) , ( 0 , - 2 )$$
C) $( 0,2 )$
D) $( 0 , - 2 )$
E) none of the above

Question 18

The solution of the system $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - y$ is

A) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$
B) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$
C) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$
D) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$
E) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$

Question 19

The geometric configuration of the solutions of $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 y$ in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 20

Consider the differential equation $x ^ { \prime } = \sin x + \cos x$ . The point $x = 3 \pi / 4$ is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 21

The geometric configuration of the solutions of $\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 y$ in the phase plane is

A) degenerate stable node
B) degenerate unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 22

The differential equation $x ^ { \prime \prime } - x ^ { \prime } e ^ { x } = 0$ can be rewritten as the system

A) $x ^ { \prime } = x , u ^ { \prime } = x e ^ { x }$
B) $x ^ { \prime } = x , u ^ { \prime } = u e ^ { x }$
C) $x ^ { \prime } = u , u ^ { \prime } = x e ^ { u }$
D) $x ^ { \prime } = u , u ^ { \prime } = x e ^ { x }$
E) $x ^ { \prime } = u , u ^ { \prime } = u e ^ { x }$

Question 23

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0,2 )$

A) $A = \left[ \begin{array} { c c } 0 & - 4 \\5 & 0\end{array} \right]$
B) $A = \left[ \begin{array} { l l } 0 & 4 \\5 & 0\end{array} \right]$
C) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & 2 y \\5 & 0\end{array} \right]$
D) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 2 y \\5 & 0\end{array} \right]$
E) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 2 y \\0 & 5\end{array} \right]$

Question 24

The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 y$ is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 25

Consider the differential equation $x ^ { \prime } = \tan x$ . The point $x = 0$ is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 26

The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 y$ is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 27

Consider the differential equation $x ^ { \prime } = \cot x$ . The point $x = \pi / 2$ is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 28

The solution of the system $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 y$ is

A) $x = 3 c _ { 1 } e ^ { - 3 t } + c _ { 2 } e ^ { - t } , y = - 5 c _ { 1 } e ^ { - 3 t } - c _ { 2 } e ^ { - t }$
B) $x = 3 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { - t } , y = - 5 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { - t }$
C) $x = 3 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { t } , y = - 5 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { t }$
D) $x = 5 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { t } , y = - 3 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { t }$
E) $x = 5 c _ { 1 } e ^ { - 3 t } + c _ { 2 } e ^ { - t } , y = - 3 c _ { 1 } e ^ { - 3 t } - c _ { 2 } e ^ { - t }$

Question 29

The values of $c$ that make the system $\frac { d x } { d t } = 3 x + 2 y , \frac { d y } { d t } = - c x + y$ stable are

A) $c > - 3 / 2$
B) $c < - 3 / 2$
C) It is stable for all values of $c$ .
D) It is unstable for all values of $c$ .
E) $c > 0$

Question 30

The geometric configuration of the solutions of $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 y$ in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 31

The critical points of the system $\frac { d x } { d t } = 2 x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 y$ are

A) $y = 0 , y = - 3$
B) $x = 0 , x = - 9 / 2$
C) $$( 0,0 )$$
D) $( 0,0 ) , ( - 9 / 2 , - 3 )$
E) $( - 9 / 2 , - 3 )$

Question 32

Which of the following systems are autonomous? Select all that apply.

A) $\frac { d x } { d t } = x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 y$
B) $\frac { d x } { d t } = x + t , \frac { d y } { d t } = 2 x - 3 y$
C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x - 3 y$
D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \sin x - 3 y$
E) $\frac { d x } { d t } = t + 1 , \frac { d y } { d t } = 15 x - 4 y$

Question 33

Which of the following systems are linear? Select all that apply.

A) $\frac { d x } { d t } = x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 y$
B) $\frac { d x } { d t } = x + t , \frac { d y } { d t } = 2 x - 3 y$
C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x - 3 y$
D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \sin x - 3 y$
E) $\frac { d x } { d t } = t + 1 , \frac { d y } { d t } = 15 x - 4 y$

Question 34

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0 , - 2 )$

A) $A = \left[ \begin{array} { c c } 0 & - 4 \\5 & 0\end{array} \right]$
B) $A = \left[ \begin{array} { l l } 0 & 4 \\5 & 0\end{array} \right]$
C) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & 2 y \\5 & 0\end{array} \right]$
D) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 2 y \\5 & 0\end{array} \right]$
E) $A = \left[ \begin{array} { r r } 3 x ^ { 2 } & - 2 y \\0 & 5\end{array} \right]$

Question 35

The critical point $( 0,2 )$ of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ is a

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) center point

Question 36

The solution of the system $\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 y$ is

A) $x = c _ { 1 } e ^ { - 3 t } + c _ { 2 } ( t + 1 ) e ^ { - 3 t } , y = c _ { 1 } e ^ { - 3 t } + c _ { 2 } t e ^ { - 3 t }$
B) $x = c _ { 1 } e ^ { - 3 t } + c _ { 2 } t e ^ { - 3 t } , y = c _ { 1 } e ^ { - 3 t } + c _ { 2 } ( t + 1 ) e ^ { - 3 t }$
C) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } t e ^ { 3 t } , y = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t }$
D) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t } , y = c _ { 1 } e ^ { 3 t } + c _ { 2 } t e ^ { 3 t }$
E) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t } , y = - c _ { 1 } e ^ { 3 t } - c _ { 2 } t e ^ { 3 t }$

Question 37

The critical point $( 0 , - 2 )$ of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ is a

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 38

The only constant solution of $\frac { d x } { d t } = 5 x - y - 4 , \frac { d y } { d t } = x + 2 y - 3$ is

A) $x = 1 , y = 1$
B) $x = 1$
C) $x = - 1$
D) $x = - 1 , y = - 1$
E) $y = 1$

Question 39

Assume that $x ( t )$ and $y ( t )$ represent the populations of two competing species at time $t$ . The Lotka-Volterra competition model is

A) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } + x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$
B) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x + a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$
C) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$
D) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } + y - a _ { 21 } y \right) / K _ { 2 }$
E) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y + a _ { 21 } y \right) / K _ { 2 }$

Question 40

The critical points of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 5 x$ are

A) $( 2,0 ) , ( 0,2 )$
B) $( - 2,0 ) , ( 0 , - 2 )$
C) $( 0,2 )$
D) $( 0 , - 2 )$
E) none of the above