Deck 2: First-Order Differential Equations

Question 1

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

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

Answer 1

1.2331

Question 2

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

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

Answer 2

linear

Question 3

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

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) $2 x + 4 y + 5 = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$

Answer 3

$2 x + 4 y + 5 = \tan ( 2 \sqrt { 2 } x + c ) / \sqrt { 2 }$

Question 4

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

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 5

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

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

Question 6

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

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 7

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

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 8

The differential equation $( x - 2 y ) d x + y d y = 0$ can be solved using the 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 9

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

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

Question 10

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

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 11

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

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

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

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

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 14

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

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

Question 15

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

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

Question 16

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

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 17

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

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

Question 18

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

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

Question 19

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

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 20

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

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

Question 21

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

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

Question 22

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

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 23

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

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

Question 24

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

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

Question 25

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

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

Question 26

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

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 27

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

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 28

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

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

Question 29

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

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

Question 30

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

A) increasing everywhere
B) decreasing everywhere
C) increasing if $0 < x < 1$
D) decreasing if $- 1 < x < 0$
E) increasing if $x > 1$

Question 31

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

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 32

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

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 33

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

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

Question 34

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

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

Question 35

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

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

Question 36

The differential equation $( x + 2 y ) d x + y d y = 0$ can be solved using the 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 37

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

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 38

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

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 39

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

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 40

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

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