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&#10;A) stable node&#10;B) unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) saddle point

Accepted Answer

The system can be represented in matrix form as $\frac{d}{dt} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -3 & 2 \ 4 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$. The eigenvalues of the matrix determine the nature of the phase plane. For this matrix, the eigenvalues are real and have opposite signs, indicating a saddle point, which is characterized by trajectories that approach the origin along one axis and diverge along another.

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 }

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 }

Accepted Answer

The correct differential equation for the horizontal position of the bead, considering both the gravitational force component along the curve and the damping force, is given by choice C. This equation accounts for the gravitational force component acting along the slope of the curve, $ - m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) $, and the damping force, $ - \beta \frac { d x } { d t } $, which opposes the direction of velocity.

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&#10;A) $c > - 3 / 2$&#10;B) $c < - 3 / 2$&#10;C) It is locally stable for all values of $C$ .&#10;D) It is unstable for all values of $C$ .&#10;E) $c > 0$

Accepted Answer

To determine the stability of the system, we can analyze the eigenvalues of the coefficient matrix of the system. The system can be represented in matrix form as $\frac{d}{dt} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -3 & 2 \ -c & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$. The eigenvalues $\lambda$ of this matrix are found by solving the characteristic equation $\det(A - \lambda I) = 0$, where $A = \begin{pmatrix} -3 & 2 \ -c & -1 \end{pmatrix}$ and $I$ is the identity matrix. This leads to the equation $\lambda^2 + 4\lambda + (3 + 2c) = 0$. For the system to be stable, all eigenvalues must have negative real parts, which requires the constant term $3 + 2c > 0$, simplifying to $c > -3/2$.

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&#10;A) $x = 2 / 3 , y = 1 / 3$&#10;B) $x = 2 / 3 , y = 1 / 3$ and $x = - 2 / 3 , y = - 1 / 3$&#10;C) $x = - 2 / 3 , y = - 1 / 3$&#10;D) $x = 2 / \sqrt { 7 } , y = 1 / \sqrt { 7 }$&#10;E) $x = 2 / \sqrt { 7 } , y = 1 / \sqrt { 7 }$ and $x = - 2 / \sqrt { 7 } , y = - 1 / \sqrt { 7 }$

Accepted Answer

For constant solutions, both derivatives must be zero. Setting $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} = 0$, we solve the system of equations $2x^2 + y^2 - 1 = 0$ and $x - 2y = 0$. Solving these simultaneously gives two sets of solutions: $x = 2/3, y = 1/3$ and $x = -2/3, y = -1/3$.

Question 5

Which of the following systems are linear? Select all that apply.&#10;A) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 t - 3 \sin t$&#10;B) $\frac { d x } { d t } = 3.5 x + 2 y , \frac { d y } { d t } = 2 x - 5 y$&#10;C) $\frac { d x } { d t } = x + 1 / y , \frac { d y } { d t } = 2 x e ^ { t } - 3 y$&#10;D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \cos x - 3 y$&#10;E) $\frac { d x } { d t } = t ^ { 2 } + 1 , \frac { d y } { d t } = 15 x - 4 y$

Accepted Answer

The answer of Which of the following systems are linear?...

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.&#10;A) asymptotically stable&#10;B) stable but not asymptotically stable&#10;C) unstable&#10;D) an attractor&#10;E) a repeller

Accepted Answer

The critical point $(0,0)$ is asymptotically stable and an attractor if the real parts of all eigenvalues of the system's Jacobian matrix at that point are negative. For the given system, the Jacobian matrix at $(0,0)$ is $\begin{pmatrix} -6 & -5 \ 4 & 2 \end{pmatrix}$. The eigenvalues of this matrix can be found by solving the characteristic equation $\det(J - \lambda I) = 0$, where $J$ is the Jacobian matrix and $I$ is the identity matrix. The characteristic equation is $(-\lambda + 2)(-\lambda - 6) - (-5)(4) = \lambda^2 + 4\lambda - 8 - 20 = \lambda^2 + 4\lambda - 28 = 0$. The solutions to this equation, or the eigenvalues, are found using the quadratic formula, yielding real parts that are negative. Thus, the critical point is asymptotically stable and acts as an attractor, since trajectories in its vicinity will converge to it over time.

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.&#10;A) asymptotically stable&#10;B) stable but not asymptotically stable&#10;C) unstable&#10;D) an attractor&#10;E) a repeller

Accepted Answer

The stability of a critical point can be determined by analyzing the eigenvalues of the system's Jacobian matrix at that point. For the given system, the Jacobian matrix is $\begin{pmatrix} 5 & 1 \ 3 & 3 \end{pmatrix}$. The eigenvalues of this matrix are found by solving the characteristic equation $\det(J - \lambda I) = 0$, where $J$ is the Jacobian matrix and $I$ is the identity matrix. This leads to the equation $(5 - \lambda)(3 - \lambda) - 3 = \lambda^2 - 8\lambda + 12 = 0$. Solving this quadratic equation for $\lambda$ gives two eigenvalues. Without explicitly solving it here, if both eigenvalues have positive real parts, the critical point is unstable and acts as a repeller. Given the coefficients of the system, it's clear that the real parts of the eigenvalues are positive (since the trace of the matrix, which is the sum of the eigenvalues, is positive and greater than the determinant, which implies that the eigenvalues themselves must be positive), indicating that the system is indeed unstable and acts as 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&#10;A) $x = c _ { 1 } e ^ { - 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) + 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$&#10;B) $x = c _ { 1 } e ^ { 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { 2 t } \sin ( 2 t )$&#10;C) $x = c _ { 1 } e ^ { - 2 t } ( 4 \cos ( 2 t ) + 2 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( - 2 \cos ( 2 t ) + 4 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$&#10;D) $x = c _ { 1 } e ^ { - 2 t } ( 2 \cos ( 2 t ) + 4 \sin ( 2 t ) ) + c _ { 2 } e ^ { - 2 t } ( 4 \cos ( 2 t ) - 2 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { - 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { - 2 t } \sin ( 2 t )$&#10;E) $x = c _ { 1 } e ^ { 2 t } ( 4 \cos ( 2 t ) + 2 \sin ( 2 t ) ) + c _ { 2 } e ^ { 2 t } ( - 2 \cos ( 2 t ) + 4 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { 2 t } \sin ( 2 t )$

Accepted Answer

The answer of The solution of the system \(\frac {...

Question 9

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 8 y$ at the critical point $( 2,0 )$&#10;A) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 3 y ^ { 2 } \&#10;8 & 0&#10;\end{array} \right]$&#10;B) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 3 y ^ { 2 } \&#10;0 & 8&#10;\end{array} \right]$&#10;C) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & 3 y ^ { 2 } \&#10;0 & 8&#10;\end{array} \right]$&#10;D) $A = \left[ \begin{array} { c c } &#10;12 & 0 \&#10;0 & 8&#10;\end{array} \right]$&#10;E) $A = \left[ \begin{array} { c c } &#10;12 & 0 \&#10;8 & 0&#10;\end{array} \right]$

Accepted Answer

The answer of The Jacobian matrix of the system \(x...

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&#10;A) $x = c _ { 1 } e ^ { - 6 t } + c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { - 6 t } - 3 c _ { 2 } e ^ { - 2 t }$&#10;B) $x = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { 2 t } , y = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { 2 t }$&#10;C) $x = c _ { 1 } e ^ { - 6 t } - 3 c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { - 6 t } + c _ { 2 } e ^ { - 2 t }$&#10;D) $x = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { - 2 t }$&#10;E) $x = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { - 2 t }$

Accepted Answer

The answer of The solution of the system \(\frac {...

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&#10;A) $x ^ { \prime } = x , u ^ { \prime } = u - t x , x ( 0 ) = 1 , u ( 0 ) = 2$&#10;B) $x ^ { \prime } = x , u ^ { \prime } = x - t u , x ( 0 ) = 1 , u ( 0 ) = 2$&#10;C) $x ^ { \prime } = u , u ^ { \prime } = u + t x , x ( 0 ) = 1 , u ( 0 ) = 2$&#10;D) $x ^ { \prime } = u , u ^ { \prime } = x + t u , x ( 0 ) = 1 , u ( 0 ) = 2$&#10;E) $x ^ { \prime } = u , u ^ { \prime } = u - t x , x ( 0 ) = 1 , u ( 0 ) = 2$

Accepted Answer

The answer of The initial value problem \(x ^ {...

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&#10;A) stable spiral point&#10;B) unstable spiral point&#10;C) stable node&#10;D) unstable node&#10;E) saddle point

Accepted Answer

The answer of The geometric configuration of the solutions of...

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&#10;A) $y = - 1$&#10;B) $x = 2$&#10;C) $( 2 , - 1 )$&#10;D) $( - 1,2 )$&#10;E) $( - 3 , - 7 )$

Accepted Answer

The answer of The critical points of the system \(\frac...

Question 14

Consider the differential equation $x ^ { \prime } = \sec x$ . The point $x = 0$ is&#10;A) a stable critical point&#10;B) an unstable critical point&#10;C) not a critical point

Accepted Answer

The answer of Consider the differential equation \(x ^ {...

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.&#10;A) asymptotically stable&#10;B) stable but not asymptotically stable&#10;C) unstable&#10;D) an attractor&#10;E) a repeller

Accepted Answer

The answer of The critical point $( 0,0 )$ of...

Question 16

Which of the following systems are autonomous? Select all that apply.&#10;A) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 t - 3 \sin t$&#10;B) $$\frac { d x } { d t } = 3.5 x + 2 y , \frac { d y } { d t } = 2 x - 5 y$$&#10;C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x e ^ { t } - 3 y$&#10;D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \cos x - 3 y$&#10;E) $\frac { d x } { d t } = t ^ { 2 } + 1 , \frac { d y } { d t } = 15 x - 4 y$

Accepted Answer

The answer of Which of the following systems are autonomous?...

Question 17

The critical points of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } - 8 , y ^ { \prime } = 8 y$ are&#10;A) $( 2,0 ) , 0,2 )$&#10;B) $$( - 2,0 ) , ( 0 , - 2 )$$&#10;C) $( 0,2 )$&#10;D) $( 0 , - 2 )$&#10;E) none of the above

Accepted Answer

The answer of The critical points of the system \(x...

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&#10;A) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$&#10;B) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$&#10;C) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$&#10;D) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$&#10;E) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$

Accepted Answer

The answer of The solution of the system \(\frac {...

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&#10;A) stable node&#10;B) unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) saddle point

Accepted Answer

The answer of The geometric configuration of the solutions of...

Question 20

Consider the differential equation $x ^ { \prime } = \sin x + \cos x$ . The point $x = 3 \pi / 4$ is&#10;A) a stable critical point&#10;B) an unstable critical point&#10;C) not a critical point

Accepted Answer

The answer of Consider the differential equation \(x ^ {...

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&#10;A) degenerate stable node&#10;B) degenerate unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) saddle point

Accepted Answer

The answer of The geometric configuration of the solutions of...

Question 22

The differential equation $x ^ { \prime \prime } - x ^ { \prime } e ^ { x } = 0$ can be rewritten as the system&#10;A) $x ^ { \prime } = x , u ^ { \prime } = x e ^ { x }$&#10;B) $x ^ { \prime } = x , u ^ { \prime } = u e ^ { x }$&#10;C) $x ^ { \prime } = u , u ^ { \prime } = x e ^ { u }$&#10;D) $x ^ { \prime } = u , u ^ { \prime } = x e ^ { x }$&#10;E) $x ^ { \prime } = u , u ^ { \prime } = u e ^ { x }$

Accepted Answer

The answer of The differential equation \(x ^ { \prime...

Question 23

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0,2 )$&#10;A) $A = \left[ \begin{array} { c c } &#10;0 & - 4 \&#10;5 & 0&#10;\end{array} \right]$&#10;B) $A = \left[ \begin{array} { l l } &#10;0 & 4 \&#10;5 & 0&#10;\end{array} \right]$&#10;C) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & 2 y \&#10;5 & 0&#10;\end{array} \right]$&#10;D) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 2 y \&#10;5 & 0&#10;\end{array} \right]$&#10;E) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 2 y \&#10;0 & 5&#10;\end{array} \right]$

Accepted Answer

The answer of The Jacobian matrix of the system \(x...

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.&#10;A) asymptotically stable&#10;B) stable but not asymptotically stable&#10;C) unstable&#10;D) an attractor&#10;E) a repeller

Accepted Answer

The answer of The critical point $( 0,0 )$ of...

Question 25

Consider the differential equation $x ^ { \prime } = \tan x$ . The point $x = 0$ is&#10;A) a stable critical point&#10;B) an unstable critical point&#10;C) not a critical point

Accepted Answer

The answer of Consider the differential equation \(x ^ {...

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.&#10;A) asymptotically stable&#10;B) stable but not asymptotically stable&#10;C) unstable&#10;D) an attractor&#10;E) a repeller

Accepted Answer

The answer of The critical point $( 0,0 )$ of...

Question 27

Consider the differential equation $x ^ { \prime } = \cot x$ . The point $x = \pi / 2$ is&#10;A) a stable critical point&#10;B) an unstable critical point&#10;C) not a critical point

Accepted Answer

The answer of Consider the differential equation \(x ^ {...

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&#10;A) $x = 3 c _ { 1 } e ^ { - 3 t } + c _ { 2 } e ^ { - t } , y = - 5 c _ { 1 } e ^ { - 3 t } - c _ { 2 } e ^ { - t }$&#10;B) $x = 3 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { - t } , y = - 5 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { - t }$&#10;C) $x = 3 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { t } , y = - 5 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { t }$&#10;D) $x = 5 c _ { 1 } e ^ { 3 t } + c _ { 2 } e ^ { t } , y = - 3 c _ { 1 } e ^ { 3 t } - c _ { 2 } e ^ { t }$&#10;E) $x = 5 c _ { 1 } e ^ { - 3 t } + c _ { 2 } e ^ { - t } , y = - 3 c _ { 1 } e ^ { - 3 t } - c _ { 2 } e ^ { - t }$

Accepted Answer

The answer of The solution of the system \(\frac {...

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&#10;A) $c > - 3 / 2$&#10;B) $c < - 3 / 2$&#10;C) It is stable for all values of $c$ .&#10;D) It is unstable for all values of $c$ .&#10;E) $c > 0$

Accepted Answer

The answer of The values of $c$ that make the...

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&#10;A) stable node&#10;B) unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) saddle point

Accepted Answer

The answer of The geometric configuration of the solutions of...

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&#10;A) $y = 0 , y = - 3$&#10;B) $x = 0 , x = - 9 / 2$&#10;C) $$( 0,0 )$$&#10;D) $( 0,0 ) , ( - 9 / 2 , - 3 )$&#10;E) $( - 9 / 2 , - 3 )$

Accepted Answer

The answer of The critical points of the system \(\frac...

Question 32

Which of the following systems are autonomous? Select all that apply.&#10;A) $\frac { d x } { d t } = x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 y$&#10;B) $\frac { d x } { d t } = x + t , \frac { d y } { d t } = 2 x - 3 y$&#10;C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x - 3 y$&#10;D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \sin x - 3 y$&#10;E) $\frac { d x } { d t } = t + 1 , \frac { d y } { d t } = 15 x - 4 y$

Accepted Answer

The answer of Which of the following systems are autonomous?...

Question 33

Which of the following systems are linear? Select all that apply.&#10;A) $\frac { d x } { d t } = x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 y$&#10;B) $\frac { d x } { d t } = x + t , \frac { d y } { d t } = 2 x - 3 y$&#10;C) $\frac { d x } { d t } = x + y , \frac { d y } { d t } = 2 x - 3 y$&#10;D) $\frac { d x } { d t } = 0 , \frac { d y } { d t } = 2 \sin x - 3 y$&#10;E) $\frac { d x } { d t } = t + 1 , \frac { d y } { d t } = 15 x - 4 y$

Accepted Answer

The answer of Which of the following systems are linear?...

Question 34

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ at the critical point $( 0 , - 2 )$&#10;A) $A = \left[ \begin{array} { c c } &#10;0 & - 4 \&#10;5 & 0&#10;\end{array} \right]$&#10;B) $A = \left[ \begin{array} { l l } &#10;0 & 4 \&#10;5 & 0&#10;\end{array} \right]$&#10;C) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & 2 y \&#10;5 & 0&#10;\end{array} \right]$&#10;D) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 2 y \&#10;5 & 0&#10;\end{array} \right]$&#10;E) $A = \left[ \begin{array} { r r } &#10;3 x ^ { 2 } & - 2 y \&#10;0 & 5&#10;\end{array} \right]$

Accepted Answer

The answer of The Jacobian matrix of the system \(x...

Question 35

The critical point $( 0,2 )$ of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ is a&#10;A) stable node&#10;B) unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) center point

Accepted Answer

The answer of The critical point $( 0,2 )$ of...

Question 36

The solution of the system $\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 y$ is&#10;A) $x = c _ { 1 } e ^ { - 3 t } + c _ { 2 } ( t + 1 ) e ^ { - 3 t } , y = c _ { 1 } e ^ { - 3 t } + c _ { 2 } t e ^ { - 3 t }$&#10;B) $x = c _ { 1 } e ^ { - 3 t } + c _ { 2 } t e ^ { - 3 t } , y = c _ { 1 } e ^ { - 3 t } + c _ { 2 } ( t + 1 ) e ^ { - 3 t }$&#10;C) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } t e ^ { 3 t } , y = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t }$&#10;D) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t } , y = c _ { 1 } e ^ { 3 t } + c _ { 2 } t e ^ { 3 t }$&#10;E) $x = c _ { 1 } e ^ { 3 t } + c _ { 2 } ( t + 1 ) e ^ { 3 t } , y = - c _ { 1 } e ^ { 3 t } - c _ { 2 } t e ^ { 3 t }$

Accepted Answer

The answer of The solution of the system \(\frac {...

Question 37

The critical point $( 0 , - 2 )$ of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$ is a&#10;A) stable node&#10;B) unstable node&#10;C) stable spiral point&#10;D) unstable spiral point&#10;E) saddle point

Accepted Answer

The answer of The critical point \(( 0 , -...

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&#10;A) $x = 1 , y = 1$&#10;B) $x = 1$&#10;C) $x = - 1$&#10;D) $x = - 1 , y = - 1$&#10;E) $y = 1$

Accepted Answer

The answer of The only constant solution of \(\frac {...

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&#10;A) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } + x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$&#10;B) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x + a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$&#10;C) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y - a _ { 21 } y \right) / K _ { 2 }$&#10;D) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } + y - a _ { 21 } y \right) / K _ { 2 }$&#10;E) $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y + a _ { 21 } y \right) / K _ { 2 }$

Accepted Answer

The answer of Assume that $x ( t )$ and...

Question 40

The critical points of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 5 x$ are&#10;A) $( 2,0 ) , ( 0,2 )$&#10;B) $( - 2,0 ) , ( 0 , - 2 )$&#10;C) $( 0,2 )$&#10;D) $( 0 , - 2 )$&#10;E) none of the above

Accepted Answer

The answer of The critical points of the system \(x...

Deck 10: Plane Autonomous Systems