Question 1

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.

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 2

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.

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 3

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.

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 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

Accepted Answer

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 $x = - 2 / \sqrt { 7 } , y = - 1 / \sqrt { 7 }$ 
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

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Which of the following systems are autonomous? 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 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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)  $x ^ { \prime } = u , u ^ { \prime } = u - t x , x ( 0 ) = 1 , u ( 0 ) = 2$ 
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 10

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

Accepted Answer

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)  $A = \left[ \begin{array} { c c } 
12 & 0 \
8 & 0
\end{array} \right]$ 
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 11

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

Accepted Answer

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

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

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

Accepted Answer

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)  $x ^ { \prime } = u , u ^ { \prime } = u e ^ { x }$ 
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 14

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

Accepted Answer

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

Question 15

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

Question 16

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

Accepted Answer

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

Question 17

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

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 18

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)  $c  0$

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Exam 10: Plane Autonomous Systems

The critical point (0,0)( 0,0 )(0,0) 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 Select all that apply.

The critical point (0,0)( 0,0 )(0,0) 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 Select all that apply.

The critical point (0,0)( 0,0 )(0,0) 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 Select all that apply.

The constant solution of dxdt=2x2+y2−1,dydt=x−2y\frac { d x } { d t } = 2 x ^ { 2 } + y ^ { 2 } - 1 , \frac { d y } { d t } = x - 2 ydtdx​=2x2+y2−1,dtdy​=x−2y are

The critical points of the system x′=x3−y3−8,y′=5xx ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 5 xx′=x3−y3−8,y′=5x are

Consider the differential equation x′=cot⁡xx ^ { \prime } = \cot xx′=cotx . The point x=π/2x = \pi / 2x=π/2 is

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

Consider the differential equation x′=sec⁡xx ^ { \prime } = \sec xx′=secx . The point x=0x = 0x=0 is

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

The Jacobian matrix of the system x′=x3−y3−8,y′=8yx ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 8 yx′=x3−y3−8,y′=8y at the critical point (2,0)( 2,0 )(2,0)

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)

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

Consider the differential equation x′=sin⁡x+cos⁡xx ^ { \prime } = \sin x + \cos xx′=sinx+cosx . The point x=3π/4x = 3 \pi / 4x=3π/4 is

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

Consider the differential equation x′=tan⁡xx ^ { \prime } = \tan xx′=tanx . The point x=0x = 0x=0 is

The geometric configuration of the solutions of 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 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

The only constant solution of dxdt=5x−y−4,dydt=x+2y−3\frac { d x } { d t } = 5 x - y - 4 , \frac { d y } { d t } = x + 2 y - 3dtdx​=5x−y−4,dtdy​=x+2y−3 is

The critical points of the system x′=x3−y2−8,y′=8yx ^ { \prime } = x ^ { 3 } - y ^ { 2 } - 8 , y ^ { \prime } = 8 yx′=x3−y2−8,y′=8y 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 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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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