Question 1

The general solution of the first-order differential equation $x ^ { 2 } y ^ { \prime } + ( 1 - 2 x ) y = x ^ { 2 }$ is

Accepted Answer

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

Question 2

The general solution of $( y + 3 ) d x + \cot x d y = 0$ is

Accepted Answer

A)  $y ^ { 2 } = c \cos x - 3$ 
B)  $3 y = c \cos x + 3$ 
C)  $3 y = c \cos x - 3$ 
D)  $y = c \cos x - 3$ 
E)  $y = c \cos x + 3$ 
A)  $y ^ { 2 } = c \cos x - 3$ 
B)  $3 y = c \cos x + 3$ 
C)  $3 y = c \cos x - 3$ 
D)  $y = c \cos x - 3$ 
E)  $y = c \cos x + 3$

Question 3

The general solution of the first-order differential equation $( x + 1 ) y ^ { \prime } - 2 y = ( x + 1 ) ^ { 4 }$ is

Accepted Answer

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

Question 4

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

Accepted Answer

A)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
B)  $y = A \left( 1 + x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
C)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( - x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
D)  $y = A \left( 1 + x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
E)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
A)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
B)  $y = A \left( 1 + x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
C)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( - x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
D)  $y = A \left( 1 + x ^ { 2 } \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 120 } - \cdots \right)$ 
E)  $y = A \left( 1 - x ^ { 2 } \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 } - \cdots \right)$

Question 5

Which of the following is a homogeneous function?

Accepted Answer

A)  $f ( x , y ) = \cos ( x y ) f ( x , y ) = \cos ( x y )$ 
B)  $f ( x , y ) = 2 x ^ { 5 } - 3 x ^ { 4 } y - y ^ { 5 }$ 
C)  $f ( x , y ) = \sin ( x y ) - x$ 
D)  $f ( x , y ) = 2 x - x ^ { 2 } y ^ { 3 } + y ^ { 2 }$ 
E)  $f ( x , y ) = x ^ { 2 } - x ^ { 2 } y ^ { 3 } + y ^ { 2 }$ 
A)  $f ( x , y ) = \cos ( x y ) f ( x , y ) = \cos ( x y )$ 
B)  $f ( x , y ) = 2 x ^ { 5 } - 3 x ^ { 4 } y - y ^ { 5 }$ 
C)  $f ( x , y ) = \sin ( x y ) - x$ 
D)  $f ( x , y ) = 2 x - x ^ { 2 } y ^ { 3 } + y ^ { 2 }$ 
E)  $f ( x , y ) = x ^ { 2 } - x ^ { 2 } y ^ { 3 } + y ^ { 2 }$

Question 6

A solution of $y \left( y ^ { \prime } \right) ^ { 2 } + 2 x y ^ { \prime } - y = 0$ is

Accepted Answer

A)  $4 y ^ { 2 } = ( x + 1 )$ 
B)  $y ^ { 2 } = - 4 ( x - 1 )$ 
C)  $y ^ { 2 } = - 4 ( x + 1 )$ 
D)  $y ^ { 2 } = 4 ( x + 1 )$ 
E)  $y ^ { 2 } = 2 ( x - 1 )$ 
A)  $4 y ^ { 2 } = ( x + 1 )$ 
B)  $y ^ { 2 } = - 4 ( x - 1 )$ 
C)  $y ^ { 2 } = - 4 ( x + 1 )$ 
D)  $y ^ { 2 } = 4 ( x + 1 )$ 
E)  $y ^ { 2 } = 2 ( x - 1 )$

Question 7

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

Accepted Answer

A)  $x ^ { 2 } - x y - 2 c x + c ^ { 2 } = 0$ 
B)  $x ^ { 2 } - x y + 2 c x + c ^ { 2 } = 0$ 
C)  $x ^ { 2 } + x y + 2 c x + c ^ { 2 } = 0$ 
D)  $x ^ { 2 } + x y - 2 c x + c ^ { 2 } = 0$ 
E)  $2 x ^ { 2 } - x y + 2 c x + c ^ { 2 } = 0$ 
A)  $x ^ { 2 } - x y - 2 c x + c ^ { 2 } = 0$ 
B)  $x ^ { 2 } - x y + 2 c x + c ^ { 2 } = 0$ 
C)  $x ^ { 2 } + x y + 2 c x + c ^ { 2 } = 0$ 
D)  $x ^ { 2 } + x y - 2 c x + c ^ { 2 } = 0$ 
E)  $2 x ^ { 2 } - x y + 2 c x + c ^ { 2 } = 0$

Question 8

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

Accepted Answer

A)  $y ^ { 2 } \sin \left( \frac { y } { x } \right) = C$ 
B)  $x ^ { 2 } \cos \left( \frac { x } { y } \right) = C$ 
C)  $x ^ { 2 } \sin \left( \frac { y } { x } \right) = C$ 
D)  $x ^ { 2 } \cos \left( \frac { y } { x } \right) = C$ 
E)  $x ^ { 2 } \sin \left( \frac { x } { y } \right) = C$ 
A)  $y ^ { 2 } \sin \left( \frac { y } { x } \right) = C$ 
B)  $x ^ { 2 } \cos \left( \frac { x } { y } \right) = C$ 
C)  $x ^ { 2 } \sin \left( \frac { y } { x } \right) = C$ 
D)  $x ^ { 2 } \cos \left( \frac { y } { x } \right) = C$ 
E)  $x ^ { 2 } \sin \left( \frac { x } { y } \right) = C$

Question 9

The general solution of the exact differential equation $( 6 x - 2 y + 1 ) d x - ( 2 x - 2 y + 3 ) d y = 0$ is

Accepted Answer

A)  $3 x ^ { 2 } - 2 x v - v ^ { 2 } + x + 3 v = C$ 
B)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } + x - 3 y = C$ 
C)  $3 x ^ { 2 } - 2 x y - y ^ { 2 } + x - 3 y = C$ 
D)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } - x - 3 y = C$ 
E)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } + x + 3 y = C$ 
A)  $3 x ^ { 2 } - 2 x v - v ^ { 2 } + x + 3 v = C$ 
B)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } + x - 3 y = C$ 
C)  $3 x ^ { 2 } - 2 x y - y ^ { 2 } + x - 3 y = C$ 
D)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } - x - 3 y = C$ 
E)  $3 x ^ { 2 } - 2 x y + y ^ { 2 } + x + 3 y = C$

Question 10

A solution of $y ^ { \prime } = x ^ { 2 } y ^ { 3 }$ is

Accepted Answer

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

Question 11

Using Taylor series about c = 1, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime } + 2 ( x - 1 ) y = 0$ is

Accepted Answer

A)  $y = A \left( 1 + ( x - 1 ) ^ { 2 } - \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$ 
B)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } - \cdots \right)$ 
C)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } - \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } - \cdots \right)$ 
D)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$ 
E)  $y = A \left( 1 + ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } + \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$ 
A)  $y = A \left( 1 + ( x - 1 ) ^ { 2 } - \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$ 
B)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } - \cdots \right)$ 
C)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } - \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } - \cdots \right)$ 
D)  $y = A \left( 1 - ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } - \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$ 
E)  $y = A \left( 1 + ( x - 1 ) ^ { 2 } + \frac { ( x - 1 ) ^ { 4 } } { 2 } + \frac { ( x - 1 ) ^ { 6 } } { 6 } + \cdots \right)$

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $\sqrt { \frac { x } { y } } - \ln y = C$ 
B)  $\sqrt { \frac { x } { y } } + \ln y = C$ 
C)  $\sqrt { \frac { x } { y } } + \ln x = C$ 
D)  $\sqrt { \frac { x } { y } } - \ln x = C$ 
E)  $- \sqrt { \frac { x } { y } } + \ln y = C$ 
A)  $\sqrt { \frac { x } { y } } - \ln y = C$ 
B)  $\sqrt { \frac { x } { y } } + \ln y = C$ 
C)  $\sqrt { \frac { x } { y } } + \ln x = C$ 
D)  $\sqrt { \frac { x } { y } } - \ln x = C$ 
E)  $- \sqrt { \frac { x } { y } } + \ln y = C$

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

A)  $\cos y = 1 + C e ^ { - \cot x }$ 
B)  $\sin y = 1 + C e ^ { - \sin x }$ 
C)  $\cos y = 1 + C e ^ { - \sin x }$ 
D)  $\sin y = 1 + C e ^ { - \cos x }$ 
E)  $\cos y = 1 + C e ^ { - \cos x }$ 
A)  $\cos y = 1 + C e ^ { - \cot x }$ 
B)  $\sin y = 1 + C e ^ { - \sin x }$ 
C)  $\cos y = 1 + C e ^ { - \sin x }$ 
D)  $\sin y = 1 + C e ^ { - \cos x }$ 
E)  $\cos y = 1 + C e ^ { - \cos x }$

Question 17

The general solution of the first-order differential equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$ is

Accepted Answer

A)  $4 x y = e ^ { x } + C e ^ { - x }$ 
B)  $3 x y = e ^ { x } - C e ^ { - x }$ 
C)  $3 x y = e ^ { x } + C e ^ { - x }$ 
D)  $2 x y = e ^ { x } - C e ^ { - x }$ 
E)  $2 x y = e ^ { x } + C e ^ { - x }$ 
A)  $4 x y = e ^ { x } + C e ^ { - x }$ 
B)  $3 x y = e ^ { x } - C e ^ { - x }$ 
C)  $3 x y = e ^ { x } + C e ^ { - x }$ 
D)  $2 x y = e ^ { x } - C e ^ { - x }$ 
E)  $2 x y = e ^ { x } + C e ^ { - x }$

Question 18

Which of the following is an exact differential equation?

Accepted Answer

A)  $( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y$ 
B)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cot x ) d y$ 
C)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sec y - \cos x ) d y$ 
D)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$ 
E)  $( y \sec x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$ 
A)  $( y \sin x - 3 \cot y ) d x + ( 3 x \sin y - \cos x ) d y$ 
B)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cot x ) d y$ 
C)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sec y - \cos x ) d y$ 
D)  $( y \sin x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$ 
E)  $( y \sec x - 3 \cos y ) d x + ( 3 x \sin y - \cos x ) d y$

Question 19

Which of the following is a homogeneous function?

Accepted Answer

A)  $f ( x , y ) = - x ^ { 2 } - x ^ { 3 } y + y ^ { 2 }$ 
B)  $f ( x , y ) = x ^ { 2 } + x y - x ^ { 3 } y ^ { 2 }$ 
C)  $f ( x , y ) = x ^ { 2 } - x ^ { 3 } y - y ^ { 2 }$ 
D)  $f ( x , y ) = x ^ { 2 } + x y + x ^ { 3 } y ^ { 2 }$ 
E)  $f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 }$ 
A)  $f ( x , y ) = - x ^ { 2 } - x ^ { 3 } y + y ^ { 2 }$ 
B)  $f ( x , y ) = x ^ { 2 } + x y - x ^ { 3 } y ^ { 2 }$ 
C)  $f ( x , y ) = x ^ { 2 } - x ^ { 3 } y - y ^ { 2 }$ 
D)  $f ( x , y ) = x ^ { 2 } + x y + x ^ { 3 } y ^ { 2 }$ 
E)  $f ( x , y ) = x ^ { 2 } - x y + y ^ { 2 }$

Question 20

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

Accepted Answer

A)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x + \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x + \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
C)  $y = A \left( 1 + \frac { x ^ { 3 } } { 3 ! } - \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } + \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } - \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } + \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
A)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x + \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x + \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
C)  $y = A \left( 1 + \frac { x ^ { 3 } } { 3 ! } - \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } + \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } - \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } - \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 3 ! } + \frac { 4 ^ { 2 } x ^ { 6 } } { 6 ! } + \frac { 4 ^ { 2 } \cdot 7 ^ { 2 } x ^ { 9 } } { 9 ! } + \cdots \right) + B \left( x - \frac { 2 ^ { 2 } x ^ { 4 } } { 4 ! } + \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } x ^ { 7 } } { 7 ! } - \frac { 2 ^ { 2 } \cdot 5 ^ { 2 } \cdot 8 ^ { 2 } x ^ { 10 } } { 10 ! } + \cdots \right)$

The general solution of the first-order differential equation $x ^ { 2 } y ^ { \prime } + ( 1 - 2 x ) y = x ^ { 2 }$ is

The general solution of $( y + 3 ) d x + \cot x d y = 0$ is

The general solution of the first-order differential equation $( x + 1 ) y ^ { \prime } - 2 y = ( x + 1 ) ^ { 4 }$ is

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

Which of the following is a homogeneous function?

A solution of $y \left( y ^ { \prime } \right) ^ { 2 } + 2 x y ^ { \prime } - y = 0$ is

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

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

The general solution of the exact differential equation $( 6 x - 2 y + 1 ) d x - ( 2 x - 2 y + 3 ) d y = 0$ is

A solution of $y ^ { \prime } = x ^ { 2 } y ^ { 3 }$ is

Using Taylor series about c = 1, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime } + 2 ( x - 1 ) y = 0$ is

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

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

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

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

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

The general solution of the first-order differential equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$ is

Which of the following is an exact differential equation?

Which of the following is a homogeneous function?

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

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Exam 17: Differential Equations

The general solution of the first-order differential equation x2y′+(1−2x)y=x2x ^ { 2 } y ^ { \prime } + ( 1 - 2 x ) y = x ^ { 2 }x2y′+(1−2x)y=x2 is

The general solution of (y+3)dx+cot⁡xdy=0( y + 3 ) d x + \cot x d y = 0(y+3)dx+cotxdy=0 is

The general solution of the first-order differential equation (x+1)y′−2y=(x+1)4( x + 1 ) y ^ { \prime } - 2 y = ( x + 1 ) ^ { 4 }(x+1)y′−2y=(x+1)4 is

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

Which of the following is a homogeneous function?

A solution of y(y′)2+2xy′−y=0y \left( y ^ { \prime } \right) ^ { 2 } + 2 x y ^ { \prime } - y = 0y(y′)2+2xy′−y=0 is

The general solution of xy′+y=2xyx y ^ { \prime } + y = 2 \sqrt { x y }xy′+y=2xy​ is

The general solution of the exact differential equation (6x−2y+1)dx−(2x−2y+3)dy=0( 6 x - 2 y + 1 ) d x - ( 2 x - 2 y + 3 ) d y = 0(6x−2y+1)dx−(2x−2y+3)dy=0 is

A solution of y′=x2y3y ^ { \prime } = x ^ { 2 } y ^ { 3 }y′=x2y3 is

Using Taylor series about c = 1, the general series solution, with the first three nonzero terms, of the differential equation y′+2(x−1)y=0y ^ { \prime } + 2 ( x - 1 ) y = 0y′+2(x−1)y=0 is

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

The general solution of the first-order differential equation yy′+xy2=x3y y ^ { \prime } + x y ^ { 2 } = x ^ { 3 }yy′+xy2=x3 is

The general solution of the exact differential equation dxxy+(2y−xy3)dy=0\frac { d x } { \sqrt { x y } } + \left( \frac { 2 } { y } - \sqrt { \frac { x } { y ^ { 3 } } } \right) d y = 0xy​dx​+(y2​−y3x​​)dy=0 is

The general solution of y2dx−(xy+x2)dy=0y ^ { 2 } d x - \left( x y + x ^ { 2 } \right) d y = 0y2dx−(xy+x2)dy=0 is

The general solution of the first-order differential equation sin⁡yy′+sin⁡xcos⁡y=sin⁡x\sin y y ^ { \prime } + \sin x \cos y = \sin xsinyy′+sinxcosy=sinx is

The general solution of the first-order differential equation xy′+(1+x)y=exx y ^ { \prime } + ( 1 + x ) y = e ^ { x }xy′+(1+x)y=ex is

Which of the following is an exact differential equation?

Which of the following is a homogeneous function?

Using Maclaurin series, the general series solution, with the first four nonzero terms, of the differential equation y′′+x2y′+xy=0y ^ { \prime \prime } + x ^ { 2 } y ^ { \prime } + x y = 0y′′+x2y′+xy=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 $x ^ { 2 } y ^ { \prime } + ( 1 - 2 x ) y = x ^ { 2 }$ is

The general solution of $( y + 3 ) d x + \cot x d y = 0$ is

The general solution of the first-order differential equation $( x + 1 ) y ^ { \prime } - 2 y = ( x + 1 ) ^ { 4 }$ is

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

A solution of $y \left( y ^ { \prime } \right) ^ { 2 } + 2 x y ^ { \prime } - y = 0$ is

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

The general solution of the exact differential equation $( 6 x - 2 y + 1 ) d x - ( 2 x - 2 y + 3 ) d y = 0$ is

A solution of $y ^ { \prime } = x ^ { 2 } y ^ { 3 }$ is

Using Taylor series about c = 1, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime } + 2 ( x - 1 ) y = 0$ is

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

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

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

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

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

The general solution of the first-order differential equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$ is

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