Question 1

The general solution of the first-order differential equation $y ^ { \prime } + y \cot x = 3 e ^ { \cos x }$ is

Accepted Answer

A)  $y \cos x = 3 e ^ { \cos x } + C$ 
B)  $y \sin x = 3 e ^ { \cos x } + C$ 
C)  $y \cos x = - 3 e ^ { \sin x } + C$ 
D)  $y \sin x = - 3 e ^ { \sin x } + C$ 
E)  $y \sin x = - 3 e ^ { \cos x } + C$ 
A)  $y \cos x = 3 e ^ { \cos x } + C$ 
B)  $y \sin x = 3 e ^ { \cos x } + C$ 
C)  $y \cos x = - 3 e ^ { \sin x } + C$ 
D)  $y \sin x = - 3 e ^ { \sin x } + C$ 
E)  $y \sin x = - 3 e ^ { \cos x } + C$ E

Question 2

The general solution of the first-order differential equation $y ^ { \prime } + ( \cot x ) y = \sec x$ is

Accepted Answer

A)  $y \sin x = \ln ( \operatorname { cosc } x ) + C$ 
B)  $y \cos x = \ln ( \sec x ) + C$ 
C)  $y \sin x = \ln ( \sec x ) + C$ 
D)  $y \cos x = \ln ( \cos x ) + C$ 
E)  $y \sec x = \ln ( \sin x ) + C$ 
A)  $y \sin x = \ln ( \operatorname { cosc } x ) + C$ 
B)  $y \cos x = \ln ( \sec x ) + C$ 
C)  $y \sin x = \ln ( \sec x ) + C$ 
D)  $y \cos x = \ln ( \cos x ) + C$ 
E)  $y \sec x = \ln ( \sin x ) + C$ C

Question 3

A solution of $\left( y ^ { \prime } \right) ^ { 2 } + y ^ { 2 } = 1$ is

Accepted Answer

A)  $y = \cos ( x - 4 )$ 
B)  $y = \cos ( x + 4 )$ 
C)  $y = - \sin ( x + 4 )$ 
D)  $y = \sin ( x - 4 )$ 
E) All of the above 
A)  $y = \cos ( x - 4 )$ 
B)  $y = \cos ( x + 4 )$ 
C)  $y = - \sin ( x + 4 )$ 
D)  $y = \sin ( x - 4 )$ 
E) All of the above E

Question 4

The general solution of the exact differential equation $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y = 0$ is

Accepted Answer

A)  $- y \ln ( 2 x - 2 ) + \ln y = C$ 
B)  $y \ln ( 2 x + 2 ) - \ln y = C$ 
C)  $y \ln ( 2 x - 2 ) - \ln y = C$ 
D)  $y \ln ( 2 x - 2 ) + \ln y = C$ 
E)  $y \ln ( 2 x + 2 ) + \ln y = C$ 
A)  $- y \ln ( 2 x - 2 ) + \ln y = C$ 
B)  $y \ln ( 2 x + 2 ) - \ln y = C$ 
C)  $y \ln ( 2 x - 2 ) - \ln y = C$ 
D)  $y \ln ( 2 x - 2 ) + \ln y = C$ 
E)  $y \ln ( 2 x + 2 ) + \ln y = C$

Question 5

A solution of $y ^ { \prime \prime } + y = 0$ is

Accepted Answer

A)  $y = \sin x + 2 \cos ^ { 2 } x$ 
B)  $y = 2 \sin ^ { 2 } x - \cos x$ 
C)  $y = - 2 \sin x + \cos ^ { 2 } x$ 
D)  $y = 2 \sin x - 2 \cos ^ { 2 } x$ 
E)  $y = 2 \sin x + \cos x$ 
A)  $y = \sin x + 2 \cos ^ { 2 } x$ 
B)  $y = 2 \sin ^ { 2 } x - \cos x$ 
C)  $y = - 2 \sin x + \cos ^ { 2 } x$ 
D)  $y = 2 \sin x - 2 \cos ^ { 2 } x$ 
E)  $y = 2 \sin x + \cos x$

Question 6

The general solution of the first-order differential equation $\left( x ^ { 3 } + x \right) y ^ { \prime } + 4 x ^ { 2 } y = 2$ is

Accepted Answer

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

Question 7

Using Taylor series about c = 2, the general series solution, with the first three nonzero terms, of the differential equation $x ^ { 2 } y ^ { \prime } - y ^ { \prime } + y = 0$ is

Accepted Answer

A)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right) + B \left( ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$ 
C)  $y = A \left( 1 + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$ 
E)  $y = A \left( 1 + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$ 
A)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right) + B \left( ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$ 
C)  $y = A \left( 1 + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$ 
E)  $y = A \left( 1 + \frac { ( x - 2 ) ^ { 2 } } { 8 } + \frac { 5 ( x - 2 ) ^ { 3 } } { 96 } + \cdots \right) + B \left( ( x - 2 ) - \frac { ( x - 2 ) ^ { 2 } } { 8 } - \frac { ( x - 2 ) ^ { 3 } } { 96 } - \cdots \right)$

Question 8

The general solution of the exact differential equation $\sqrt { x y } \left[ \left( 5 x y - 3 y ^ { 3 } \right) d x + \left( 3 x ^ { 2 } - 7 x y ^ { 2 } \right) \right] = 0$ is

Accepted Answer

A)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
B)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } + x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
C)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
D)  $x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - x ^ { \frac { 3 } { 2 } } y ^ { \frac { 7 } { 2 } } = C$ 
E)  $x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } + x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
A)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
B)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } + x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
C)  $2 x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$ 
D)  $x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } - x ^ { \frac { 3 } { 2 } } y ^ { \frac { 7 } { 2 } } = C$ 
E)  $x ^ { \frac { 5 } { 2 } } y ^ { \frac { 3 } { 2 } } + x ^ { \frac { 3 } { 2 } } y ^ { \frac { 5 } { 2 } } = C$

Question 9

A solution of $y ^ { \prime \prime } - 7 y ^ { \prime } + 12 y = 0$ is

Accepted Answer

A)  $y = 2 e ^ { 3 x } + 5 e ^ { 4 x }$ 
B)  $y = 2 e ^ { 3 x } - 5 e ^ { 4 x }$ 
C)  $y = - 2 e ^ { 3 x } + 5 e ^ { 4 x }$ 
D)  $y = - 2 e ^ { 3 x } - 5 e ^ { 4 x }$ 
E) All of the above 
A)  $y = 2 e ^ { 3 x } + 5 e ^ { 4 x }$ 
B)  $y = 2 e ^ { 3 x } - 5 e ^ { 4 x }$ 
C)  $y = - 2 e ^ { 3 x } + 5 e ^ { 4 x }$ 
D)  $y = - 2 e ^ { 3 x } - 5 e ^ { 4 x }$ 
E) All of the above

Question 10

Which of the following is an exact differential equation?

Accepted Answer

A)  $\left( 3 x ^ { 2 } y + x ^ { 3 } y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y ^ { 3 } + \cos y \right) d y$ 
B)  $\left( 3 x ^ { 2 } y - x y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y + \cos y \right) d y$ 
C)  $\left( 3 x ^ { 2 } y - x y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } - x ^ { 2 } y + \cos y \right) d y$ 
D)  $\left( 3 x ^ { 2 } y - x y + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y + \cos y \right) d y$ 
E)  $\left( 3 x ^ { 2 } y + x y + e ^ { x } \right) d x + \left( x ^ { 4 } - x ^ { 2 } y + \cos y \right) d y$ 
A)  $\left( 3 x ^ { 2 } y + x ^ { 3 } y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y ^ { 3 } + \cos y \right) d y$ 
B)  $\left( 3 x ^ { 2 } y - x y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y + \cos y \right) d y$ 
C)  $\left( 3 x ^ { 2 } y - x y ^ { 2 } + e ^ { x } \right) d x + \left( x ^ { 3 } - x ^ { 2 } y + \cos y \right) d y$ 
D)  $\left( 3 x ^ { 2 } y - x y + e ^ { x } \right) d x + \left( x ^ { 3 } + x ^ { 2 } y + \cos y \right) d y$ 
E)  $\left( 3 x ^ { 2 } y + x y + e ^ { x } \right) d x + \left( x ^ { 4 } - x ^ { 2 } y + \cos y \right) d y$

Question 11

A solution of $y ^ { \prime \prime } - y = 0$ is

Accepted Answer

A)  $y = e ^ { x } + e ^ { - x }$ 
B)  $y = e ^ { x } - e ^ { - x }$ 
C)  $y = - e ^ { x } + e ^ { - x }$ 
D)  $y = - e ^ { x } - e ^ { - x }$ 
E) All of the above 
A)  $y = e ^ { x } + e ^ { - x }$ 
B)  $y = e ^ { x } - e ^ { - x }$ 
C)  $y = - e ^ { x } + e ^ { - x }$ 
D)  $y = - e ^ { x } - e ^ { - x }$ 
E) All of the above

Question 12

The general solution of $x y ^ { \prime } - y + \sqrt { y ^ { 2 } - x ^ { 2 } } = 0$ is

Accepted Answer

A)  $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { 2 } = c x ^ { 3 }$ 
B)  $\left( x ^ { 2 } - y ^ { 2 } \right) y ^ { 2 } = c x ^ { 3 }$ 
C)  $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { 2 } = c x ^ { 4 }$ 
D)  $\left( x ^ { 2 } - y ^ { 2 } \right) y ^ { 2 } = c x ^ { 4 }$ 
E)  $\left( x ^ { 2 } + y ^ { 2 } \right) y = c x ^ { 4 }$ 
A)  $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { 2 } = c x ^ { 3 }$ 
B)  $\left( x ^ { 2 } - y ^ { 2 } \right) y ^ { 2 } = c x ^ { 3 }$ 
C)  $\left( x ^ { 2 } + y ^ { 2 } \right) y ^ { 2 } = c x ^ { 4 }$ 
D)  $\left( x ^ { 2 } - y ^ { 2 } \right) y ^ { 2 } = c x ^ { 4 }$ 
E)  $\left( x ^ { 2 } + y ^ { 2 } \right) y = c x ^ { 4 }$

Question 13

The general solution of the first-order differential equation $y d x - \left( 3 x + y ^ { 4 } \right) d y = 0$ is

Accepted Answer

A)  $x = y ^ { 3 } + C y ^ { 3 }$ 
B)  $x = y ^ { 3 } - C y ^ { 4 }$ 
C)  $x = y ^ { 3 } + C y ^ { 4 }$ 
D)  $x = y ^ { 4 } + C y ^ { 3 }$ 
E)  $x = - y ^ { 4 } + C y ^ { 3 }$ 
A)  $x = y ^ { 3 } + C y ^ { 3 }$ 
B)  $x = y ^ { 3 } - C y ^ { 4 }$ 
C)  $x = y ^ { 3 } + C y ^ { 4 }$ 
D)  $x = y ^ { 4 } + C y ^ { 3 }$ 
E)  $x = - y ^ { 4 } + C y ^ { 3 }$

Question 14

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + x ^ { 2 } y ^ { \prime } + x y = 0$ is

Accepted Answer

A)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } + \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
A)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 45 } + \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 6 } + \frac { 5 x ^ { 7 } } { 252 } + \cdots \right)$

Question 15

Which of the following is an exact differential equation?

Accepted Answer

A)  $( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y$ 
B)  $( x + y ) d x + ( 2 y + x ) d y$ 
C)  $\left( x ^ { 2 } + 2 x y \right) d x + ( 2 y + x ) d y$ 
D)  $( x + y ) d x + \left( 2 y ^ { 2 } + 3 x y \right) d y$ 
E)  $\left( x + y ^ { 2 } \right) d x + ( 2 y + x ) d y$ 
A)  $( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y$ 
B)  $( x + y ) d x + ( 2 y + x ) d y$ 
C)  $\left( x ^ { 2 } + 2 x y \right) d x + ( 2 y + x ) d y$ 
D)  $( x + y ) d x + \left( 2 y ^ { 2 } + 3 x y \right) d y$ 
E)  $\left( x + y ^ { 2 } \right) d x + ( 2 y + x ) d y$

Question 16

The general solution of the exact differential equation $\left( \frac { x ^ { 2 } } { y ^ { 2 } } \cos \left( \frac { x } { y } \right) \right) \frac { d y } { d x } = \frac { x } { y } \cos \left( \frac { x } { y } \right) + \sin \left( \frac { x } { y } \right) + \cos x$ is

Accepted Answer

A)  $- x \sin \left( \frac { y } { x } \right) - \sin x = C$ 
B)  $x \sin \left( \frac { x } { y } \right) + \sin x = C$ 
C)  $- x \sin \left( \frac { x } { y } \right) - \sin x = C$ 
D)  $- x \sin \left( \frac { x } { y } \right) + \sin x = C$ 
E)  $x \sin \left( \frac { x } { y } \right) - \sin x = C$ 
A)  $- x \sin \left( \frac { y } { x } \right) - \sin x = C$ 
B)  $x \sin \left( \frac { x } { y } \right) + \sin x = C$ 
C)  $- x \sin \left( \frac { x } { y } \right) - \sin x = C$ 
D)  $- x \sin \left( \frac { x } { y } \right) + \sin x = C$ 
E)  $x \sin \left( \frac { x } { y } \right) - \sin x = C$

Question 17

The general solution of $x y d x - \left( 1 + x ^ { 2 } \right) d y = 0$ is

Accepted Answer

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

Question 18

A solution of $x y ^ { \prime \prime } - y ^ { \prime } = 0$ is

Accepted Answer

A)  $y = x ^ { 2 } + 3$ 
B)  $y = - x ^ { 2 } - 2$ 
C)  $y = x ^ { 2 } + 2$ 
D)  $y = x ^ { 2 } - 2$ 
E) All of the above 
A)  $y = x ^ { 2 } + 3$ 
B)  $y = - x ^ { 2 } - 2$ 
C)  $y = x ^ { 2 } + 2$ 
D)  $y = x ^ { 2 } - 2$ 
E) All of the above

Question 19

The general solution of the Bernoulli equation $y ^ { \prime } + \left( \frac { x } { 1 - x ^ { 2 } } \right) y = x y ^ { \frac { 1 } { 2 } }$ is

Accepted Answer

A)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { - 4 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
B)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { - 2 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
C)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 4 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
D)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 2 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
E)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 3 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
A)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { - 4 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
B)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { - 2 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
C)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 4 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
D)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 2 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$ 
E)  $\sqrt { y } = \frac { x ^ { 2 } - 1 } { 3 } + C \left( x ^ { 2 } - 1 \right) ^ { \frac { 1 } { 4 } }$

Question 20

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + 2 x ^ { 2 } y = 0$ is

Accepted Answer

A)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x + \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } + \cdots \right)$ 
B)  $A \left( 1 + \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } + \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } - \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } - \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
A)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x + \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } + \cdots \right)$ 
B)  $A \left( 1 + \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } + \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } + \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } - \frac { x ^ { 9 } } { 360 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 6 } - \frac { x ^ { 8 } } { 168 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 10 } + \frac { x ^ { 9 } } { 360 } - \cdots \right)$

The general solution of the first-order differential equation $y ^ { \prime } + y \cot x = 3 e ^ { \cos x }$ is

The general solution of the first-order differential equation $y ^ { \prime } + ( \cot x ) y = \sec x$ is

A solution of $\left( y ^ { \prime } \right) ^ { 2 } + y ^ { 2 } = 1$ is

The general solution of the exact differential equation $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y = 0$ is

A solution of $y ^ { \prime \prime } + y = 0$ is

The general solution of the first-order differential equation $\left( x ^ { 3 } + x \right) y ^ { \prime } + 4 x ^ { 2 } y = 2$ is

Using Taylor series about c = 2, the general series solution, with the first three nonzero terms, of the differential equation $x ^ { 2 } y ^ { \prime } - y ^ { \prime } + y = 0$ is

The general solution of the exact differential equation $\sqrt { x y } \left[ \left( 5 x y - 3 y ^ { 3 } \right) d x + \left( 3 x ^ { 2 } - 7 x y ^ { 2 } \right) \right] = 0$ is

A solution of $y ^ { \prime \prime } - 7 y ^ { \prime } + 12 y = 0$ is

Which of the following is an exact differential equation?

A solution of $y ^ { \prime \prime } - y = 0$ is

The general solution of $x y ^ { \prime } - y + \sqrt { y ^ { 2 } - x ^ { 2 } } = 0$ is

The general solution of the first-order differential equation $y d x - \left( 3 x + y ^ { 4 } \right) d y = 0$ is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + x ^ { 2 } y ^ { \prime } + x y = 0$ is

Which of the following is an exact differential equation?

The general solution of the exact differential equation $\left( \frac { x ^ { 2 } } { y ^ { 2 } } \cos \left( \frac { x } { y } \right) \right) \frac { d y } { d x } = \frac { x } { y } \cos \left( \frac { x } { y } \right) + \sin \left( \frac { x } { y } \right) + \cos x$ is

The general solution of $x y d x - \left( 1 + x ^ { 2 } \right) d y = 0$ is

A solution of $x y ^ { \prime \prime } - y ^ { \prime } = 0$ is

The general solution of the Bernoulli equation $y ^ { \prime } + \left( \frac { x } { 1 - x ^ { 2 } } \right) y = x y ^ { \frac { 1 } { 2 } }$ is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + 2 x ^ { 2 } y = 0$ is

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Exam 17: Differential Equations

The general solution of the first-order differential equation y′+ycot⁡x=3ecos⁡xy ^ { \prime } + y \cot x = 3 e ^ { \cos x }y′+ycotx=3ecosx is

The general solution of the first-order differential equation y′+(cot⁡x)y=sec⁡xy ^ { \prime } + ( \cot x ) y = \sec xy′+(cotx)y=secx is

A solution of (y′)2+y2=1\left( y ^ { \prime } \right) ^ { 2 } + y ^ { 2 } = 1(y′)2+y2=1 is

The general solution of the exact differential equation (yx−1)dx+(ln⁡(2x−2)+1y)dy=0\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y = 0(x−1y​)dx+(ln(2x−2)+y1​)dy=0 is

A solution of y′′+y=0y ^ { \prime \prime } + y = 0y′′+y=0 is

The general solution of the first-order differential equation (x3+x)y′+4x2y=2\left( x ^ { 3 } + x \right) y ^ { \prime } + 4 x ^ { 2 } y = 2(x3+x)y′+4x2y=2 is

Using Taylor series about c = 2, the general series solution, with the first three nonzero terms, of the differential equation x2y′−y′+y=0x ^ { 2 } y ^ { \prime } - y ^ { \prime } + y = 0x2y′−y′+y=0 is

The general solution of the exact differential equation xy[(5xy−3y3)dx+(3x2−7xy2)]=0\sqrt { x y } \left[ \left( 5 x y - 3 y ^ { 3 } \right) d x + \left( 3 x ^ { 2 } - 7 x y ^ { 2 } \right) \right] = 0xy​[(5xy−3y3)dx+(3x2−7xy2)]=0 is

A solution of y′′−7y′+12y=0y ^ { \prime \prime } - 7 y ^ { \prime } + 12 y = 0y′′−7y′+12y=0 is

Which of the following is an exact differential equation?

A solution of y′′−y=0y ^ { \prime \prime } - y = 0y′′−y=0 is

The general solution of xy′−y+y2−x2=0x y ^ { \prime } - y + \sqrt { y ^ { 2 } - x ^ { 2 } } = 0xy′−y+y2−x2​=0 is

The general solution of the first-order differential equation ydx−(3x+y4)dy=0y d x - \left( 3 x + y ^ { 4 } \right) d y = 0ydx−(3x+y4)dy=0 is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation y′′+x2y′+xy=0y ^ { \prime \prime } + x ^ { 2 } y ^ { \prime } + x y = 0y′′+x2y′+xy=0 is

Which of the following is an exact differential equation?

The general solution of xydx−(1+x2)dy=0x y d x - \left( 1 + x ^ { 2 } \right) d y = 0xydx−(1+x2)dy=0 is

A solution of xy′′−y′=0x y ^ { \prime \prime } - y ^ { \prime } = 0xy′′−y′=0 is

The general solution of the Bernoulli equation y′+(x1−x2)y=xy12y ^ { \prime } + \left( \frac { x } { 1 - x ^ { 2 } } \right) y = x y ^ { \frac { 1 } { 2 } }y′+(1−x2x​)y=xy21​ is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation y′′+2x2y=0y ^ { \prime \prime } + 2 x ^ { 2 } y = 0y′′+2x2y=0 is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Filters

The general solution of the first-order differential equation $y ^ { \prime } + y \cot x = 3 e ^ { \cos x }$ is

The general solution of the first-order differential equation $y ^ { \prime } + ( \cot x ) y = \sec x$ is

A solution of $\left( y ^ { \prime } \right) ^ { 2 } + y ^ { 2 } = 1$ is

The general solution of the exact differential equation $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y = 0$ is

A solution of $y ^ { \prime \prime } + y = 0$ is

The general solution of the first-order differential equation $\left( x ^ { 3 } + x \right) y ^ { \prime } + 4 x ^ { 2 } y = 2$ is

Using Taylor series about c = 2, the general series solution, with the first three nonzero terms, of the differential equation $x ^ { 2 } y ^ { \prime } - y ^ { \prime } + y = 0$ is

The general solution of the exact differential equation $\sqrt { x y } \left[ \left( 5 x y - 3 y ^ { 3 } \right) d x + \left( 3 x ^ { 2 } - 7 x y ^ { 2 } \right) \right] = 0$ is

A solution of $y ^ { \prime \prime } - 7 y ^ { \prime } + 12 y = 0$ is

A solution of $y ^ { \prime \prime } - y = 0$ is

The general solution of $x y ^ { \prime } - y + \sqrt { y ^ { 2 } - x ^ { 2 } } = 0$ is

The general solution of the first-order differential equation $y d x - \left( 3 x + y ^ { 4 } \right) d y = 0$ is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + x ^ { 2 } y ^ { \prime } + x y = 0$ is

The general solution of $x y d x - \left( 1 + x ^ { 2 } \right) d y = 0$ is

A solution of $x y ^ { \prime \prime } - y ^ { \prime } = 0$ is

The general solution of the Bernoulli equation $y ^ { \prime } + \left( \frac { x } { 1 - x ^ { 2 } } \right) y = x y ^ { \frac { 1 } { 2 } }$ is

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + 2 x ^ { 2 } y = 0$ is