Question 1

The general solution of the first-order differential equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$ is

Accepted Answer

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

Question 2

The order and degree of the differential equation $\frac { d y } { d x } = - \frac { x } { y }$ are

Accepted Answer

A) 1;1 
B) 1;2 
C) 2;1 
D) 2;2 
E) 1;3 
A) 1;1 
B) 1;2 
C) 2;1 
D) 2;2 
E) 1;3

Question 3

The general solution of $\sin x \cos ^ { 2 } y d x + \cos ^ { 2 } x d y = 0$ is

Accepted Answer

A)  $\sec x + \tan y = C$ 
B)  $\sec x - \tan y = C$ 
C)  $\sec x + \sec y = C$ 
D)  $\tan x + \tan y = C$ 
E)  $\tan x + \sec y = C$ 
A)  $\sec x + \tan y = C$ 
B)  $\sec x - \tan y = C$ 
C)  $\sec x + \sec y = C$ 
D)  $\tan x + \tan y = C$ 
E)  $\tan x + \sec y = C$

Question 4

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

Accepted Answer

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

Question 5

The general solution of the Bernoulli equation $4 x y ^ { \prime } + 3 y + e ^ { x } x ^ { 4 } y ^ { 5 } = 0$ is

Accepted Answer

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

Question 6

A solution of $y ^ { \prime \prime } + 9 y = 0$ is

Accepted Answer

A)  $y = 3 \sin ( 3 x )$ 
B)  $y = \sin ( 3 x )$ 
C)  $y = - \sin ( 3 x )$ 
D)  $y = \cos ( 3 x )$ 
E) All of the above 
A)  $y = 3 \sin ( 3 x )$ 
B)  $y = \sin ( 3 x )$ 
C)  $y = - \sin ( 3 x )$ 
D)  $y = \cos ( 3 x )$ 
E) All of the above

Question 7

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

Accepted Answer

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

Question 8

The general solution of the exact differential equation $\left( 2 y \cot x - 3 x ^ { 2 } \right) \frac { d y } { d x } = y ^ { 2 } \csc ^ { 2 } x + 6 x y - 2$ is

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
B)  $y = A \left( 1 + x ^ { 2 } - \frac { x ^ { 4 } } { 2 } + \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
C)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } - \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
D)  $y = A \left( 1 + x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
E)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } - \cdots \right)$ 
A)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
B)  $y = A \left( 1 + x ^ { 2 } - \frac { x ^ { 4 } } { 2 } + \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
C)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } - \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
D)  $y = A \left( 1 + x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$ 
E)  $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } - \cdots \right)$

Question 12

Which of the following is a homogeneous function?

Accepted Answer

A)  $f ( x , y ) = x ^ { 3 } - 2 x ^ { 2 } y + y ^ { 3 }$ 
B)  $f ( x , y ) = 2 x ^ { 3 } - 3 x y - y ^ { 3 }$ 
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)  $f ( x , y ) = x ^ { 3 } - 2 x ^ { 2 } y + y ^ { 3 }$ 
B)  $f ( x , y ) = 2 x ^ { 3 } - 3 x y - y ^ { 3 }$ 
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 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 24 } - \frac { x ^ { 7 } } { 112 } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } - \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } + \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
A)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 24 } - \frac { x ^ { 7 } } { 112 } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } - \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } - \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } + \frac { 7 x ^ { 6 } } { 240 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 24 } + \frac { x ^ { 7 } } { 112 } + \cdots \right)$

Question 15

The orthogonal trajectories of $x ^ { 2 } - y ^ { 2 } = c$ are

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $- x ^ { 2 } + y ^ { 2 } - 3 x = 0$ 
B)  $x ^ { 2 } - y ^ { 2 } + 3 x = 0$ 
C)  $x ^ { 2 } - y ^ { 2 } - 3 x = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } - 3 x = 0$ 
A)  $- x ^ { 2 } + y ^ { 2 } - 3 x = 0$ 
B)  $x ^ { 2 } - y ^ { 2 } + 3 x = 0$ 
C)  $x ^ { 2 } - y ^ { 2 } - 3 x = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } - 3 x = 0$

Question 19

The orthogonal trajectories of $y ^ { 2 } = c x$ are

Accepted Answer

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

The general solution of the first-order differential equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$ is

The order and degree of the differential equation $\frac { d y } { d x } = - \frac { x } { y }$ are

The general solution of $\sin x \cos ^ { 2 } y d x + \cos ^ { 2 } x d y = 0$ is

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

The general solution of the Bernoulli equation $4 x y ^ { \prime } + 3 y + e ^ { x } x ^ { 4 } y ^ { 5 } = 0$ is

A solution of $y ^ { \prime \prime } + 9 y = 0$ is

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

The general solution of the exact differential equation $\left( 2 y \cot x - 3 x ^ { 2 } \right) \frac { d y } { d x } = y ^ { 2 } \csc ^ { 2 } x + 6 x y - 2$ is

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

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

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

Which of the following is a homogeneous function?

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

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

The orthogonal trajectories of $x ^ { 2 } - y ^ { 2 } = c$ are

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

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

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

The orthogonal trajectories of $y ^ { 2 } = c x$ are

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Exam 17: Differential Equations

The general solution of the first-order differential equation (x2−1)y′−xy=x3−2x\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x(x2−1)y′−xy=x3−2x is

The order and degree of the differential equation dydx=−xy\frac { d y } { d x } = - \frac { x } { y }dxdy​=−yx​ are

The general solution of sin⁡xcos⁡2ydx+cos⁡2xdy=0\sin x \cos ^ { 2 } y d x + \cos ^ { 2 } x d y = 0sinxcos2ydx+cos2xdy=0 is

The general solution of (y2−xy)dx+x2dy=0\left( y ^ { 2 } - x y \right) d x + x ^ { 2 } d y = 0(y2−xy)dx+x2dy=0 is

The general solution of the Bernoulli equation 4xy′+3y+exx4y5=04 x y ^ { \prime } + 3 y + e ^ { x } x ^ { 4 } y ^ { 5 } = 04xy′+3y+exx4y5=0 is

A solution of y′′+9y=0y ^ { \prime \prime } + 9 y = 0y′′+9y=0 is

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

The general solution of the exact differential equation (2ycot⁡x−3x2)dydx=y2csc⁡2x+6xy−2\left( 2 y \cot x - 3 x ^ { 2 } \right) \frac { d y } { d x } = y ^ { 2 } \csc ^ { 2 } x + 6 x y - 2(2ycotx−3x2)dxdy​=y2csc2x+6xy−2 is

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

The general solution of the first-order differential equation xy′−yln⁡y=x2yx y ^ { \prime } - y \ln y = x ^ { 2 } yxy′−ylny=x2y is

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

Which of the following is a homogeneous function?

The general solution of the exact differential equation e2x(dydx+2y)=x2e ^ { 2 x } \left( \frac { d y } { d x } + 2 y \right) = x ^ { 2 }e2x(dxdy​+2y)=x2 is

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

The orthogonal trajectories of x2−y2=cx ^ { 2 } - y ^ { 2 } = cx2−y2=c are

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

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

A solution of 2x+2yy′=32 x + 2 y y ^ { \prime } = 32x+2yy′=3 is

The orthogonal trajectories of y2=cxy ^ { 2 } = c xy2=cx are

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The general solution of the first-order differential equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$ is

The order and degree of the differential equation $\frac { d y } { d x } = - \frac { x } { y }$ are

The general solution of $\sin x \cos ^ { 2 } y d x + \cos ^ { 2 } x d y = 0$ is

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

The general solution of the Bernoulli equation $4 x y ^ { \prime } + 3 y + e ^ { x } x ^ { 4 } y ^ { 5 } = 0$ is

A solution of $y ^ { \prime \prime } + 9 y = 0$ is

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

The general solution of the exact differential equation $\left( 2 y \cot x - 3 x ^ { 2 } \right) \frac { d y } { d x } = y ^ { 2 } \csc ^ { 2 } x + 6 x y - 2$ is

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

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

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

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

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

The orthogonal trajectories of $x ^ { 2 } - y ^ { 2 } = c$ are

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

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

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

The orthogonal trajectories of $y ^ { 2 } = c x$ are