Question 1

The differential equation $x y d x + \left( x ^ { 2 } + y ^ { 2 } \right) d y = 0$ is

Accepted Answer

A)  exact with solution $x ^ { 2 } y / 2 + y ^ { 3 } / 3 = c$ 
B)  exact with solution $x ^ { 2 } y / 2 + y ^ { 2 } / 2 = c$ 
C)  exact with solution $x ^ { 2 } y / 2 + y ^ { 3 } / 3 + c$ 
D)  not exact but having an integrating factor x 
E)  not exact but having an integrating factor y 
A)  exact with solution $x ^ { 2 } y / 2 + y ^ { 3 } / 3 = c$ 
B)  exact with solution $x ^ { 2 } y / 2 + y ^ { 2 } / 2 = c$ 
C)  exact with solution $x ^ { 2 } y / 2 + y ^ { 3 } / 3 + c$ 
D)  not exact but having an integrating factor x 
E)  not exact but having an integrating factor y E

Question 2

The differential equation $y ^ { \prime } = ( 2 x + 4 y + 5 ) ^ { 2 }$ has the solution

Accepted Answer

A)  $y = - ( 2 x + 3 ) ^ { 3 } / 6 + c$ 
B)  $y = ( 2 x + 4 y + 5 ) ^ { 3 } / 6 + c$ 
C)  $y = ( 2 x + 4 y + 5 ) ^ { 3 } / 3 + c$ 
D)  $y = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$ 
E)  $2 x + 4 y + 5 = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$ 
A)  $y = - ( 2 x + 3 ) ^ { 3 } / 6 + c$ 
B)  $y = ( 2 x + 4 y + 5 ) ^ { 3 } / 6 + c$ 
C)  $y = ( 2 x + 4 y + 5 ) ^ { 3 } / 3 + c$ 
D)  $y = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$ 
E)  $2 x + 4 y + 5 = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$ E

Question 3

The differential equation $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { \prime } = x y$ is

Accepted Answer

A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli 
A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli B

Question 4

The differential equation $x ^ { 2 } y ^ { \prime } = 2 x y + \cos x$ is

Accepted Answer

A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli 
A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli

Question 5

An integrating factor for the linear differential equation $y ^ { \prime } + y / x = x$ is

Accepted Answer

A)  $1 / x$ 
B)  $x$ 
C)  $1 / x ^ { 2 }$ 
D)  $x ^ { 2 }$ 
E)  $e ^ { - x }$ 
A)  $1 / x$ 
B)  $x$ 
C)  $1 / x ^ { 2 }$ 
D)  $x ^ { 2 }$ 
E)  $e ^ { - x }$

Question 6

Solve the problem $y ^ { \prime } = x y ^ { 2 } , y ( 1 ) = 1$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

Accepted Answer

A)  1.1 
B)  1.121 
C)  1.2331 
D)  1.23 
E)  1.221 
A)  1.1 
B)  1.121 
C)  1.2331 
D)  1.23 
E)  1.221

Question 7

The differential equation $( x - 2 y ) d x + y d y = 0$ can be solved using the substitution

Accepted Answer

A)  $u = x y$ 
B)  $u = y / x$ 
C)  $u = x - 2 y$ 
D)  $u = y$ 
E)  it cannot be solved using a substitution 
A)  $u = x y$ 
B)  $u = y / x$ 
C)  $u = x - 2 y$ 
D)  $u = y$ 
E)  it cannot be solved using a substitution

Question 8

Solve the problem $y ^ { \prime } = x y , y ( 1 ) = 2$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

Accepted Answer

A)  2.1 
B)  2.442 
C)  2.242 
D)  2.421 
E)  2.4 
A)  2.1 
B)  2.442 
C)  2.242 
D)  2.421 
E)  2.4

Question 9

The differential equation $y ^ { \prime } + y = x y ^ { 2 }$ is

Accepted Answer

A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli 
A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli

Question 10

An integrating factor for the linear differential equation $x y ^ { \prime } + y = x$ is

Accepted Answer

A)  0 
B)  1 
C)  $x$ 
D)  $1 / x$ 
E)  $e ^ { x }$ 
A)  0 
B)  1 
C)  $x$ 
D)  $1 / x$ 
E)  $e ^ { x }$

Question 11

The differential equation $y ^ { \prime } = x e ^ { y } / y$ is

Accepted Answer

A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli 
A)  linear 
B)  homogeneous 
C)  separable 
D)  exact 
E)  Bernoulli

Question 12

The solution of the differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ is

Accepted Answer

A)  $y = c / x - x / 2$ 
B)  $y = 1 / ( c / x - x / 2 )$ 
C)  $y = ( c x - x \ln x )$ 
D)  $y = 1 / ( c x - x \ln x )$ 
E)  $y = 1 + c e ^ { x }$ 
A)  $y = c / x - x / 2$ 
B)  $y = 1 / ( c / x - x / 2 )$ 
C)  $y = ( c x - x \ln x )$ 
D)  $y = 1 / ( c x - x \ln x )$ 
E)  $y = 1 + c e ^ { x }$

Question 13

Assume that $a > 0$ , $b > 0$ . The autonomous differential equation $\frac { d P } { d t } = P ( a + b P )$ has a solution that is

Accepted Answer

A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $- a / b < P < 0$ 
D)  decreasing if $- a / b < P < 0$ 
E)  decreasing if $P < - a / b$ 
A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $- a / b < P < 0$ 
D)  decreasing if $- a / b < P < 0$ 
E)  decreasing if $P < - a / b$

Question 14

An integrating factor for the linear differential equation $y ^ { \prime } - y / x = x$ is

Accepted Answer

A)  $x$ 
B)  $x ^ { 2 }$ 
C)  $1 / x$ 
D)  $1 / x ^ { 2 }$ 
E)  $e ^ { - x }$ 
A)  $x$ 
B)  $x ^ { 2 }$ 
C)  $1 / x$ 
D)  $1 / x ^ { 2 }$ 
E)  $e ^ { - x }$

Question 15

The autonomous differential equation $\frac { d x } { d t } = x ( x - 1 ) ( x + 1 )$ has a solution that is

Accepted Answer

A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $0  1$ 
A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $0  1$

Question 16

Solve the problem $y ^ { \prime } = x ^ { 2 } y ^ { 2 } , y ( 0 ) = 1$ numerically for $y ( 0.2 )$ using $h = 0.1$

Accepted Answer

A)  1.0 
B)  1.001 
C)  1.01 
D)  1.02 
E)  1.002 
A)  1.0 
B)  1.001 
C)  1.01 
D)  1.02 
E)  1.002

Question 17

The differential equation $y ^ { \prime } = \sqrt { 2 x - y + 1 } + 2$ has the solution

Accepted Answer

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

Question 18

In the autonomous differential equation $\frac { d x } { d t } = x ( 1 - x )$ , the critical point

Accepted Answer

A)  $x = 0$ is an attractor 
B)  $x = 0$ is semistable 
C)  $x = 1$ is an attractor 
D)  $x = 1$ is a repeller 
E)  $x = 1$ is semistable 
A)  $x = 0$ is an attractor 
B)  $x = 0$ is semistable 
C)  $x = 1$ is an attractor 
D)  $x = 1$ is a repeller 
E)  $x = 1$ is semistable

Question 19

The solution of the differential equation $y ^ { \prime } + y = x$ is

Accepted Answer

A)  $y = x - 1 + c e ^ { - x }$ 
B)  $y = x ^ { 2 } / 2 + e ^ { x }$ 
C)  $y = x ^ { 2 } / 2 + e ^ { - x }$ 
D)  $y = x - 1 + c e ^ { x }$ 
E)  $y = - x - 1 + c e ^ { x }$ 
A)  $y = x - 1 + c e ^ { - x }$ 
B)  $y = x ^ { 2 } / 2 + e ^ { x }$ 
C)  $y = x ^ { 2 } / 2 + e ^ { - x }$ 
D)  $y = x - 1 + c e ^ { x }$ 
E)  $y = - x - 1 + c e ^ { x }$

Question 20

In the autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( 1 - x )$ , the critical point

Accepted Answer

A)  $x = 0$ is an attractor 
B)  $x = 0$ is a repeller 
C)  $x = 1$ is an attractor 
D)  $x = 1$ is a repeller 
E)  $x = 1$ is semistable 
A)  $x = 0$ is an attractor 
B)  $x = 0$ is a repeller 
C)  $x = 1$ is an attractor 
D)  $x = 1$ is a repeller 
E)  $x = 1$ is semistable

The differential equation $x y d x + \left( x ^ { 2 } + y ^ { 2 } \right) d y = 0$ is

The differential equation $y ^ { \prime } = ( 2 x + 4 y + 5 ) ^ { 2 }$ has the solution

The differential equation $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { \prime } = x y$ is

The differential equation $x ^ { 2 } y ^ { \prime } = 2 x y + \cos x$ is

An integrating factor for the linear differential equation $y ^ { \prime } + y / x = x$ is

Solve the problem $y ^ { \prime } = x y ^ { 2 } , y ( 1 ) = 1$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

The differential equation $( x - 2 y ) d x + y d y = 0$ can be solved using the substitution

Solve the problem $y ^ { \prime } = x y , y ( 1 ) = 2$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

The differential equation $y ^ { \prime } + y = x y ^ { 2 }$ is

An integrating factor for the linear differential equation $x y ^ { \prime } + y = x$ is

The differential equation $y ^ { \prime } = x e ^ { y } / y$ is

The solution of the differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ is

Assume that $a > 0$ , $b > 0$ . The autonomous differential equation $\frac { d P } { d t } = P ( a + b P )$ has a solution that is

An integrating factor for the linear differential equation $y ^ { \prime } - y / x = x$ is

The autonomous differential equation $\frac { d x } { d t } = x ( x - 1 ) ( x + 1 )$ has a solution that is

Solve the problem $y ^ { \prime } = x ^ { 2 } y ^ { 2 } , y ( 0 ) = 1$ numerically for $y ( 0.2 )$ using $h = 0.1$

The differential equation $y ^ { \prime } = \sqrt { 2 x - y + 1 } + 2$ has the solution

In the autonomous differential equation $\frac { d x } { d t } = x ( 1 - x )$ , the critical point

The solution of the differential equation $y ^ { \prime } + y = x$ is

In the autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( 1 - x )$ , the critical point

Introduction to 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

Plane Autonomous Systems

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 2: First-Order Differential Equations

The differential equation xydx+(x2+y2)dy=0x y d x + \left( x ^ { 2 } + y ^ { 2 } \right) d y = 0xydx+(x2+y2)dy=0 is

The differential equation y′=(2x+4y+5)2y ^ { \prime } = ( 2 x + 4 y + 5 ) ^ { 2 }y′=(2x+4y+5)2 has the solution

The differential equation (x2+y2)y′=xy\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { \prime } = x y(x2+y2)y′=xy is

The differential equation x2y′=2xy+cos⁡xx ^ { 2 } y ^ { \prime } = 2 x y + \cos xx2y′=2xy+cosx is

An integrating factor for the linear differential equation y′+y/x=xy ^ { \prime } + y / x = xy′+y/x=x is

Solve the problem y′=xy2,y(1)=1y ^ { \prime } = x y ^ { 2 } , y ( 1 ) = 1y′=xy2,y(1)=1 numerically for y(1.2)y ( 1.2 )y(1.2) using h=0.1h = 0.1h=0.1 .

The differential equation (x−2y)dx+ydy=0( x - 2 y ) d x + y d y = 0(x−2y)dx+ydy=0 can be solved using the substitution

Solve the problem y′=xy,y(1)=2y ^ { \prime } = x y , y ( 1 ) = 2y′=xy,y(1)=2 numerically for y(1.2)y ( 1.2 )y(1.2) using h=0.1h = 0.1h=0.1 .

The differential equation y′+y=xy2y ^ { \prime } + y = x y ^ { 2 }y′+y=xy2 is

An integrating factor for the linear differential equation xy′+y=xx y ^ { \prime } + y = xxy′+y=x is

The differential equation y′=xey/yy ^ { \prime } = x e ^ { y } / yy′=xey/y is

The solution of the differential equation y′−y/x=y2y ^ { \prime } - y / x = y ^ { 2 }y′−y/x=y2 is

Assume that a>0a > 0a>0 , b>0b > 0b>0 . The autonomous differential equation dPdt=P(a+bP)\frac { d P } { d t } = P ( a + b P )dtdP​=P(a+bP) has a solution that is

An integrating factor for the linear differential equation y′−y/x=xy ^ { \prime } - y / x = xy′−y/x=x is

The autonomous differential equation dxdt=x(x−1)(x+1)\frac { d x } { d t } = x ( x - 1 ) ( x + 1 )dtdx​=x(x−1)(x+1) has a solution that is

Solve the problem y′=x2y2,y(0)=1y ^ { \prime } = x ^ { 2 } y ^ { 2 } , y ( 0 ) = 1y′=x2y2,y(0)=1 numerically for y(0.2)y ( 0.2 )y(0.2) using h=0.1h = 0.1h=0.1

The differential equation y′=2x−y+1+2y ^ { \prime } = \sqrt { 2 x - y + 1 } + 2y′=2x−y+1​+2 has the solution

In the autonomous differential equation dxdt=x(1−x)\frac { d x } { d t } = x ( 1 - x )dtdx​=x(1−x) , the critical point

The solution of the differential equation y′+y=xy ^ { \prime } + y = xy′+y=x is

In the autonomous differential equation dxdt=x2(1−x)\frac { d x } { d t } = x ^ { 2 } ( 1 - x )dtdx​=x2(1−x) , the critical point

Introduction to 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

Plane Autonomous Systems

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 differential equation $x y d x + \left( x ^ { 2 } + y ^ { 2 } \right) d y = 0$ is

The differential equation $y ^ { \prime } = ( 2 x + 4 y + 5 ) ^ { 2 }$ has the solution

The differential equation $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { \prime } = x y$ is

The differential equation $x ^ { 2 } y ^ { \prime } = 2 x y + \cos x$ is

An integrating factor for the linear differential equation $y ^ { \prime } + y / x = x$ is

Solve the problem $y ^ { \prime } = x y ^ { 2 } , y ( 1 ) = 1$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

The differential equation $( x - 2 y ) d x + y d y = 0$ can be solved using the substitution

Solve the problem $y ^ { \prime } = x y , y ( 1 ) = 2$ numerically for $y ( 1.2 )$ using $h = 0.1$ .

The differential equation $y ^ { \prime } + y = x y ^ { 2 }$ is

An integrating factor for the linear differential equation $x y ^ { \prime } + y = x$ is

The differential equation $y ^ { \prime } = x e ^ { y } / y$ is

The solution of the differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ is

Assume that $a > 0$ , $b > 0$ . The autonomous differential equation $\frac { d P } { d t } = P ( a + b P )$ has a solution that is

An integrating factor for the linear differential equation $y ^ { \prime } - y / x = x$ is

The autonomous differential equation $\frac { d x } { d t } = x ( x - 1 ) ( x + 1 )$ has a solution that is

Solve the problem $y ^ { \prime } = x ^ { 2 } y ^ { 2 } , y ( 0 ) = 1$ numerically for $y ( 0.2 )$ using $h = 0.1$

The differential equation $y ^ { \prime } = \sqrt { 2 x - y + 1 } + 2$ has the solution

In the autonomous differential equation $\frac { d x } { d t } = x ( 1 - x )$ , the critical point

The solution of the differential equation $y ^ { \prime } + y = x$ is

In the autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( 1 - x )$ , the critical point