Question 1

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

Accepted Answer

A)  1.1 
B)  1.11 
C)  1.2 
D)  1.21 
E)  1.221 
A)  1.1 
B)  1.11 
C)  1.2 
D)  1.21 
E)  1.221

Question 2

The autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( x - 4 )$ has a solution that is

Accepted Answer

A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $0  4$ 
E)  increasing if $x > 4$ 
A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $0  4$ 
E)  increasing if $x > 4$

Question 3

The solution of $( x - 2 y ) d x + y d y = 0$ is

Accepted Answer

A)  $\ln ( y - x ) - x / ( y - x ) = c$ 
B)  $\ln ( y - x ) - x / ( y - x ) + c$ 
C)  $\ln x + \ln ( y - x ) = c$ 
D)  $\ln ( ( y - x ) / x ) = c$ 
E)  it cannot be solved 
A)  $\ln ( y - x ) - x / ( y - x ) = c$ 
B)  $\ln ( y - x ) - x / ( y - x ) + c$ 
C)  $\ln x + \ln ( y - x ) = c$ 
D)  $\ln ( ( y - x ) / x ) = c$ 
E)  it cannot be solved

Question 4

The differential equation $2 x y d x + \left( x ^ { 2 } + 1 \right) d y = 0$ is

Accepted Answer

A)  exact with solution $x ^ { 2 } y + y + c$ 
B)  exact with solution $x ^ { 2 } y + y = c$ 
C)  exact with solution $2 x y + y + c$ 
D)  exact with solution $2 x y + y = c$ 
E)  not exact 
A)  exact with solution $x ^ { 2 } y + y + c$ 
B)  exact with solution $x ^ { 2 } y + y = c$ 
C)  exact with solution $2 x y + y + c$ 
D)  exact with solution $2 x y + y = c$ 
E)  not exact

Question 5

If $a > 0$ and $b > 0$ , 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 $0  a / b$ 
A)  increasing everywhere 
B)  decreasing everywhere 
C)  increasing if $0  a / b$

Question 6

The differential equation $( - x y \sin x + 2 y \cos x ) d x + 2 x \cos x d y = 0$ is

Accepted Answer

A)  exact with solution $- x y \cos x + y \sin x + 2 x y \cos x = c$ 
B)  exact with solution $- x y \cos x + y \sin x + 2 x y \cos x + c$ 
C)  exact with solution $- 2 x y \cos x + y \sin x + 2 x y \cos x = c$ 
D)  not exact but having an integrating factor xy 
E)  not exact but having an integrating factor y 
A)  exact with solution $- x y \cos x + y \sin x + 2 x y \cos x = c$ 
B)  exact with solution $- x y \cos x + y \sin x + 2 x y \cos x + c$ 
C)  exact with solution $- 2 x y \cos x + y \sin x + 2 x y \cos x = c$ 
D)  not exact but having an integrating factor xy 
E)  not exact but having an integrating factor y

Question 7

The differential equation $y ^ { \prime } = ( 4 x + 2 y + 3 ) ^ { 2 }$ has the solution

Accepted Answer

A)  $y = - ( 4 x + 3 ) ^ { 3 } / 12 + c$ 
B)  $y = ( 4 x + 2 y + 3 ) ^ { 3 } / 12 + c$ 
C)  $y = ( 4 x + 2 y + 3 ) ^ { 3 } / 3 + c$ 
D)  $y = \sqrt { 2 } \tan ( 2 \sqrt { 2 } x + c )$ 
E)  $4 x + 2 y + 3 = \sqrt { 2 } \tan ( 2 \sqrt { 2 } x + c )$ 
A)  $y = - ( 4 x + 3 ) ^ { 3 } / 12 + c$ 
B)  $y = ( 4 x + 2 y + 3 ) ^ { 3 } / 12 + c$ 
C)  $y = ( 4 x + 2 y + 3 ) ^ { 3 } / 3 + c$ 
D)  $y = \sqrt { 2 } \tan ( 2 \sqrt { 2 } x + c )$ 
E)  $4 x + 2 y + 3 = \sqrt { 2 } \tan ( 2 \sqrt { 2 } x + c )$

Question 8

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

Accepted Answer

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

Question 9

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 10

The differential equation $\left( y ^ { 3 } + 6 x y ^ { 4 } \right) d x + \left( 3 x y ^ { 2 } + 12 x ^ { 2 } y ^ { 3 } \right) d y = 0$ is

Accepted Answer

A)  exact with solution $y ^ { 4 } / 4 + 6 x y ^ { 5 } / 5 + 3 x ^ { 2 } y ^ { 2 } / 2 + 4 x ^ { 3 } y ^ { 3 } + c$ 
B)  exact with solution $y ^ { 4 } / 4 + 6 x y ^ { 5 } / 5 + 3 x ^ { 2 } y ^ { 2 } / 2 + 4 x ^ { 3 } y ^ { 3 } = c$ 
C)  exact with solution $x y ^ { 3 } + 3 x ^ { 2 } y ^ { 4 } = c$ 
D)  exact with solution $x y ^ { 3 } + 3 x ^ { 2 } y ^ { 4 } + c$ 
E)  not exact 
A)  exact with solution $y ^ { 4 } / 4 + 6 x y ^ { 5 } / 5 + 3 x ^ { 2 } y ^ { 2 } / 2 + 4 x ^ { 3 } y ^ { 3 } + c$ 
B)  exact with solution $y ^ { 4 } / 4 + 6 x y ^ { 5 } / 5 + 3 x ^ { 2 } y ^ { 2 } / 2 + 4 x ^ { 3 } y ^ { 3 } = c$ 
C)  exact with solution $x y ^ { 3 } + 3 x ^ { 2 } y ^ { 4 } = c$ 
D)  exact with solution $x y ^ { 3 } + 3 x ^ { 2 } y ^ { 4 } + c$ 
E)  not exact

Question 11

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

Accepted Answer

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

Question 12

The solution of $( x + 2 y ) d x + y d y = 0$ is

Accepted Answer

A)  $\ln x + \ln ( y + x ) = c$ 
B)  $\ln ( ( y + x ) / x ) = c$ 
C)  $\ln ( y + x ) + x / ( y + x ) = c$ 
D)  $\ln ( y + x ) + x / ( y + x ) + c$ 
E)  it cannot be solved 
A)  $\ln x + \ln ( y + x ) = c$ 
B)  $\ln ( ( y + x ) / x ) = c$ 
C)  $\ln ( y + x ) + x / ( y + x ) = c$ 
D)  $\ln ( y + x ) + x / ( y + x ) + c$ 
E)  it cannot be solved

Question 13

The differential equation $y ^ { \prime } + y / x = y ^ { 2 }$ can be solved using the substitution

Accepted Answer

A)  $u = y$ 
B)  $u = y ^ { 2 }$ 
C)  $u = y ^ { 3 }$ 
D)  $u = y ^ { - 1 }$ 
E)  $u = y ^ { - 2 }$ 
A)  $u = y$ 
B)  $u = y ^ { 2 }$ 
C)  $u = y ^ { 3 }$ 
D)  $u = y ^ { - 1 }$ 
E)  $u = y ^ { - 2 }$

Question 14

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

Accepted Answer

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

Question 15

The differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ can be solved using the substitution

Accepted Answer

A)  $u = y$ 
B)  $u = y ^ { 2 }$ 
C)  $u = y ^ { 3 }$ 
D)  $u = y ^ { - 1 }$ 
E)  $u = y ^ { - 2 }$ 
A)  $u = y$ 
B)  $u = y ^ { 2 }$ 
C)  $u = y ^ { 3 }$ 
D)  $u = y ^ { - 1 }$ 
E)  $u = y ^ { - 2 }$

Question 16

An integrating factor for the linear differential equation $x ^ { 2 } y ^ { \prime } + x y = 1$ 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 17

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

Accepted Answer

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

Question 18

The differential equation $2 x y d x + \left( x ^ { 2 } + y ^ { 3 } \right) d y = 0$ is

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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)  $y = - x - 1 + c e ^ { x }$

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

The autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( x - 4 )$ has a solution that is

The solution of $( x - 2 y ) d x + y d y = 0$ is

The differential equation $2 x y d x + \left( x ^ { 2 } + 1 \right) d y = 0$ is

If $a > 0$ and $b > 0$ , the autonomous differential equation $\frac { d P } { d t } = P ( a - b P )$ has a solution that is

The differential equation $( - x y \sin x + 2 y \cos x ) d x + 2 x \cos x d y = 0$ is

The differential equation $y ^ { \prime } = ( 4 x + 2 y + 3 ) ^ { 2 }$ has the solution

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

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

The differential equation $\left( y ^ { 3 } + 6 x y ^ { 4 } \right) d x + \left( 3 x y ^ { 2 } + 12 x ^ { 2 } y ^ { 3 } \right) d y = 0$ is

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

The solution of $( x + 2 y ) d x + y d y = 0$ is

The differential equation $y ^ { \prime } + y / x = y ^ { 2 }$ can be solved using the substitution

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

The differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ can be solved using the substitution

An integrating factor for the linear differential equation $x ^ { 2 } y ^ { \prime } + x y = 1$ is

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

The differential equation $2 x y d x + \left( x ^ { 2 } + y ^ { 3 } \right) d y = 0$ is

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

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

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

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

The autonomous differential equation dxdt=x2(x−4)\frac { d x } { d t } = x ^ { 2 } ( x - 4 )dtdx​=x2(x−4) has a solution that is

The solution of (x−2y)dx+ydy=0( x - 2 y ) d x + y d y = 0(x−2y)dx+ydy=0 is

The differential equation 2xydx+(x2+1)dy=02 x y d x + \left( x ^ { 2 } + 1 \right) d y = 02xydx+(x2+1)dy=0 is

If a>0a > 0a>0 and 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

The differential equation (−xysin⁡x+2ycos⁡x)dx+2xcos⁡xdy=0( - x y \sin x + 2 y \cos x ) d x + 2 x \cos x d y = 0(−xysinx+2ycosx)dx+2xcosxdy=0 is

The differential equation y′=(4x+2y+3)2y ^ { \prime } = ( 4 x + 2 y + 3 ) ^ { 2 }y′=(4x+2y+3)2 has the solution

The solution of the differential equation y′=x2yy ^ { \prime } = x ^ { 2 } yy′=x2y is

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

The differential equation (y3+6xy4)dx+(3xy2+12x2y3)dy=0\left( y ^ { 3 } + 6 x y ^ { 4 } \right) d x + \left( 3 x y ^ { 2 } + 12 x ^ { 2 } y ^ { 3 } \right) d y = 0(y3+6xy4)dx+(3xy2+12x2y3)dy=0 is

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

The solution of (x+2y)dx+ydy=0( x + 2 y ) d x + y d y = 0(x+2y)dx+ydy=0 is

The differential equation y′+y/x=y2y ^ { \prime } + y / x = y ^ { 2 }y′+y/x=y2 can be solved using the substitution

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

The differential equation y′−y/x=y2y ^ { \prime } - y / x = y ^ { 2 }y′−y/x=y2 can be solved using the substitution

An integrating factor for the linear differential equation x2y′+xy=1x ^ { 2 } y ^ { \prime } + x y = 1x2y′+xy=1 is

The differential equation xy′=2y+sin⁡xx y ^ { \prime } = 2 y + \sin xxy′=2y+sinx is

The differential equation 2xydx+(x2+y3)dy=02 x y d x + \left( x ^ { 2 } + y ^ { 3 } \right) d y = 02xydx+(x2+y3)dy=0 is

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

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

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

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

The autonomous differential equation $\frac { d x } { d t } = x ^ { 2 } ( x - 4 )$ has a solution that is

The solution of $( x - 2 y ) d x + y d y = 0$ is

The differential equation $2 x y d x + \left( x ^ { 2 } + 1 \right) d y = 0$ is

If $a > 0$ and $b > 0$ , the autonomous differential equation $\frac { d P } { d t } = P ( a - b P )$ has a solution that is

The differential equation $( - x y \sin x + 2 y \cos x ) d x + 2 x \cos x d y = 0$ is

The differential equation $y ^ { \prime } = ( 4 x + 2 y + 3 ) ^ { 2 }$ has the solution

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

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

The differential equation $\left( y ^ { 3 } + 6 x y ^ { 4 } \right) d x + \left( 3 x y ^ { 2 } + 12 x ^ { 2 } y ^ { 3 } \right) d y = 0$ is

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

The solution of $( x + 2 y ) d x + y d y = 0$ is

The differential equation $y ^ { \prime } + y / x = y ^ { 2 }$ can be solved using the substitution

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

The differential equation $y ^ { \prime } - y / x = y ^ { 2 }$ can be solved using the substitution

An integrating factor for the linear differential equation $x ^ { 2 } y ^ { \prime } + x y = 1$ is

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

The differential equation $2 x y d x + \left( x ^ { 2 } + y ^ { 3 } \right) d y = 0$ is

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

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