Question 1

The general solution of $\sin x \cos y d x + \cos x \sin y d y = 0$ is

Accepted Answer

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

Question 2

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

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Which of the following is a homogeneous function?

Accepted Answer

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

Question 7

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

Accepted Answer

A)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
C)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } + \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } + \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } + \cdots \right)$ 
A)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } + \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
C)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } + \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 6 } } { 180 } - \cdots \right) + B \left( x - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 7 } } { 504 } - \cdots \right)$ 
E)  $y = A \left( 1 + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 6 } } { 180 } + \cdots \right) + B \left( x + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 7 } } { 504 } + \cdots \right)$

Question 8

The general solution of the Bernoulli equation $x y ^ { \prime } + y = y ^ { 2 } \ln x$ is

Accepted Answer

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

Question 9

The general solution of $y ^ { 3 } d x + \left( x y ^ { 2 } + 1 \right) d y = 0$ is

Accepted Answer

A)  $- x y ^ { 2 } + c y - 1 = 0$ 
B)  $x y ^ { 2 } + c y + 1 = 0$ 
C)  $x y ^ { 2 } + c y - 1 = 0$ 
D)  $x y ^ { 2 } - c y - 1 = 0$ 
E)  $x y ^ { 2 } - c y + 1 = 0$ 
A)  $- x y ^ { 2 } + c y - 1 = 0$ 
B)  $x y ^ { 2 } + c y + 1 = 0$ 
C)  $x y ^ { 2 } + c y - 1 = 0$ 
D)  $x y ^ { 2 } - c y - 1 = 0$ 
E)  $x y ^ { 2 } - c y + 1 = 0$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

A solution of $y ^ { \prime } + y ^ { \prime } = 6 e ^ { 2 x }$ is

Accepted Answer

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

Question 13

The order and degree of the differential equation $\left( y - x y ^ { \prime } \right) ^ { 2 } = 4 \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right)$ 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 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

The general solution of the exact differential equation $\left( 2 x y ^ { - 3 } - 3 x ^ { 2 } \right) d x + 3 \left( 1 - x ^ { 2 } \right) y ^ { - 4 } d y = 0$ is

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

The general solution of $\sin x \cos y d x + \cos x \sin y d y = 0$ is

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

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

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

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

Which of the following is a homogeneous function?

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

The general solution of the Bernoulli equation $x y ^ { \prime } + y = y ^ { 2 } \ln x$ is

The general solution of $y ^ { 3 } d x + \left( x y ^ { 2 } + 1 \right) d y = 0$ is

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

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

A solution of $y ^ { \prime } + y ^ { \prime } = 6 e ^ { 2 x }$ is

The order and degree of the differential equation $\left( y - x y ^ { \prime } \right) ^ { 2 } = 4 \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right)$ are

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

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

The general solution of the exact differential equation $\left( 2 x y ^ { - 3 } - 3 x ^ { 2 } \right) d x + 3 \left( 1 - x ^ { 2 } \right) y ^ { - 4 } d y = 0$ is

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

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

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

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

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Exam 17: Differential Equations

The general solution of sin⁡xcos⁡ydx+cos⁡xsin⁡ydy=0\sin x \cos y d x + \cos x \sin y d y = 0sinxcosydx+cosxsinydy=0 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

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

The general solution of the exact differential equation (x−y)dx+(x+y)dyx2+y2=0\frac { ( x - y ) d x + ( x + y ) d y } { x ^ { 2 } + y ^ { 2 } } = 0x2+y2(x−y)dx+(x+y)dy​=0 is

The general solution of (x−ycos⁡(yx))dx+xcos⁡(yx)dy=0\left( x - y \cos \left( \frac { y } { x } \right) \right) d x + x \cos \left( \frac { y } { x } \right) d y = 0(x−ycos(xy​))dx+xcos(xy​)dy=0 is

Which of the following is a homogeneous function?

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

The general solution of the Bernoulli equation xy′+y=y2ln⁡xx y ^ { \prime } + y = y ^ { 2 } \ln xxy′+y=y2lnx is

The general solution of y3dx+(xy2+1)dy=0y ^ { 3 } d x + \left( x y ^ { 2 } + 1 \right) d y = 0y3dx+(xy2+1)dy=0 is

The general solution of (1+y2)dx+(1+x2)dy=0\left( 1 + y ^ { 2 } \right) d x + \left( 1 + x ^ { 2 } \right) d y = 0(1+y2)dx+(1+x2)dy=0 is

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

A solution of y′+y′=6e2xy ^ { \prime } + y ^ { \prime } = 6 e ^ { 2 x }y′+y′=6e2x is

The order and degree of the differential equation (y−xy′)2=4(1+(y′)2)\left( y - x y ^ { \prime } \right) ^ { 2 } = 4 \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right)(y−xy′)2=4(1+(y′)2) are

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

The general solution of the exact differential equation (2xy−3−3x2)dx+3(1−x2)y−4dy=0\left( 2 x y ^ { - 3 } - 3 x ^ { 2 } \right) d x + 3 \left( 1 - x ^ { 2 } \right) y ^ { - 4 } d y = 0(2xy−3−3x2)dx+3(1−x2)y−4dy=0 is

The orthogonal trajectories of x2−y2=cxx ^ { 2 } - y ^ { 2 } = c xx2−y2=cx are

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

A solution of y′+y=3e2xy ^ { \prime } + y = 3 e ^ { 2 x }y′+y=3e2x is

A solution of y′+2=xy ^ { \prime } + 2 = xy′+2=x 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 $\sin x \cos y d x + \cos x \sin y d y = 0$ is

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

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

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

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

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

The general solution of the Bernoulli equation $x y ^ { \prime } + y = y ^ { 2 } \ln x$ is

The general solution of $y ^ { 3 } d x + \left( x y ^ { 2 } + 1 \right) d y = 0$ is

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

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

A solution of $y ^ { \prime } + y ^ { \prime } = 6 e ^ { 2 x }$ is

The order and degree of the differential equation $\left( y - x y ^ { \prime } \right) ^ { 2 } = 4 \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right)$ are

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

The general solution of the exact differential equation $\left( 2 x y ^ { - 3 } - 3 x ^ { 2 } \right) d x + 3 \left( 1 - x ^ { 2 } \right) y ^ { - 4 } d y = 0$ is

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

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

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

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