Question 1

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

Accepted Answer

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

Question 2

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.

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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)  $x = c _ { 1 } e ^ { 2 t } ( 4 \cos ( 2 t ) + 2 \sin ( 2 t ) ) + c _ { 2 } e ^ { 2 t } ( - 2 \cos ( 2 t ) + 4 \sin ( 2 t ) )$ , $y = - 4 c _ { 1 } e ^ { 2 t } \cos ( 2 t ) - 4 c _ { 2 } e ^ { 2 t } \sin ( 2 t )$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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)  $\frac { d x } { d t } = r _ { 1 } x \left( K _ { 1 } - x - a _ { 12 } y \right) / K _ { 1 } , \frac { d y } { d t } = r _ { 2 } y \left( K _ { 2 } - y + a _ { 21 } y \right) / K _ { 2 }$ 
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 10

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

Accepted Answer

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)  $x = c _ { 1 } e ^ { 6 t } + c _ { 2 } e ^ { - 2 t } , y = c _ { 1 } e ^ { 6 t } - 3 c _ { 2 } e ^ { - 2 t }$ 
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 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

Accepted Answer

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

Question 12

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

Accepted Answer

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)  $A = \left[ \begin{array} { r r } 
3 x ^ { 2 } & - 2 y \
0 & 5
\end{array} \right]$ 
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 13

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

Accepted Answer

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)  $\frac { d x } { d t } = t + 1 , \frac { d y } { d t } = 15 x - 4 y$ 
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 14

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

Accepted Answer

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

Accepted Answer

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

Question 16

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

Accepted Answer

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)  $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$ 
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 17

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

Accepted Answer

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)  $x = 5 c _ { 1 } e ^ { - 3 t } + c _ { 2 } e ^ { - t } , y = - 3 c _ { 1 } e ^ { - 3 t } - c _ { 2 } e ^ { - t }$ 
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 18

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

Accepted Answer

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)  $\frac { d x } { d t } = t ^ { 2 } + 1 , \frac { d y } { d t } = 15 x - 4 y$ 
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 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

A)  $c > - 3 / 2$ 
B)  $c  0$ 
A)  $c > - 3 / 2$ 
B)  $c  0$

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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Exam 10: Plane Autonomous Systems

The solution of the system dxdt=4x−y,dydt=x+2y\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 ydtdx​=4x−y,dtdy​=x+2y is

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

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

The geometric configuration of the solutions of dxdt=4x−y,dydt=x+2y\frac { d x } { d t } = 4 x - y , \frac { d y } { d t } = x + 2 ydtdx​=4x−y,dtdy​=x+2y in the phase plane is

The solution of the system dxdt=−6x−5y,dydt=4x+2y\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 ydtdx​=−6x−5y,dtdy​=4x+2y is

The critical points of the system dxdt=2x+y−3,dydt=2x−3y−7\frac { d x } { d t } = 2 x + y - 3 , \frac { d y } { d t } = 2 x - 3 y - 7dtdx​=2x+y−3,dtdy​=2x−3y−7 are

The critical point (0,−2)( 0 , - 2 )(0,−2) of the system x′=x3−y2+4,y′=5xx ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 xx′=x3−y2+4,y′=5x is a

The geometric configuration of the solutions of dxdt=−6x−5y,dydt=3x+2y\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 ydtdx​=−6x−5y,dtdy​=3x+2y in the phase plane is

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

The solution of the system dxdt=5x+y,dydt=3x+3y\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 ydtdx​=5x+y,dtdy​=3x+3y is

The geometric configuration of the solutions of dxdt=5x+y,dydt=3x+3y\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 ydtdx​=5x+y,dtdy​=3x+3y in the phase plane is

The Jacobian matrix of the system x′=x3−y2+4,y′=5xx ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 xx′=x3−y2+4,y′=5x at the critical point (0,2)( 0,2 )(0,2)

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

The critical points of the system dxdt=2x+y2,dydt=2x−3y\frac { d x } { d t } = 2 x + y ^ { 2 } , \frac { d y } { d t } = 2 x - 3 ydtdx​=2x+y2,dtdy​=2x−3y are

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

The solution of the system dxdt=−3x+2y,dydt=4x−y\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - ydtdx​=−3x+2y,dtdy​=4x−y is

The solution of the system dxdt=−6x−5y,dydt=3x+2y\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 ydtdx​=−6x−5y,dtdy​=3x+2y is

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

The geometric configuration of the solutions of dxdt=−3x+2y,dydt=4x−y\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - ydtdx​=−3x+2y,dtdy​=4x−y in the phase plane is

The values of CCC that make the system dxdt=−3x+2y,dydt=−cx−y\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = - c x - ydtdx​=−3x+2y,dtdy​=−cx−y stable are

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

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

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.

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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