Question 1

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

Accepted Answer

A)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } - \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
D)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 8 } } { 672 } + \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x + \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } + \cdots \right)$ 
A)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
B)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } - \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
D)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 8 } } { 672 } + \cdots \right) + B \left( x - \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 8 } } { 672 } - \cdots \right) + B \left( x + \frac { x ^ { 5 } } { 20 } + \frac { x ^ { 9 } } { 1440 } + \cdots \right)$

Question 2

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

Accepted Answer

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

Question 3

The general solution of the Bernoulli equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$ is

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

The orthogonal trajectories of $2 x ^ { 2 } + 3 y ^ { 2 } = c$ are

Accepted Answer

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

Question 8

Which of the following is an exact differential equation?

Accepted Answer

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

Question 9

The order and degree of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - 5 x$ 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 10

The general solution of the Bernoulli equation $3 \left( 1 + x ^ { 2 } \right) y ^ { \prime } + 2 x y = 2 x y ^ { 4 }$ is

Accepted Answer

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

Question 11

The general solution of the exact differential equation $\left( 3 x ^ { 2 } - 6 x y \right) d x + \left( 12 y - 3 x ^ { 2 } \right) d y = 0$ is

Accepted Answer

A)  $- x ^ { 3 } - 3 x ^ { 2 } y + 6 y ^ { 2 } = C$ 
B)  $x ^ { 3 } + 3 x ^ { 2 } y - 6 y ^ { 2 } = C$ 
C)  $x ^ { 3 } + 3 x ^ { 2 } y + 6 y ^ { 2 } = C$ 
D)  $x ^ { 3 } - 3 x ^ { 2 } y - 6 y ^ { 2 } = C$ 
E)  $x ^ { 3 } - 3 x ^ { 2 } y + 6 y ^ { 2 } = C$ 
A)  $- x ^ { 3 } - 3 x ^ { 2 } y + 6 y ^ { 2 } = C$ 
B)  $x ^ { 3 } + 3 x ^ { 2 } y - 6 y ^ { 2 } = C$ 
C)  $x ^ { 3 } + 3 x ^ { 2 } y + 6 y ^ { 2 } = C$ 
D)  $x ^ { 3 } - 3 x ^ { 2 } y - 6 y ^ { 2 } = C$ 
E)  $x ^ { 3 } - 3 x ^ { 2 } y + 6 y ^ { 2 } = C$

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

A solution of $y ^ { \prime } + 2 y = 0$ is

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

The order and degree of the differential equation $4 \left( y ^ { \prime \prime } \right) ^ { 2 } = \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right) ^ { 3 }$ 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 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
B)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 40 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
A)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x - \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
B)  $y = A \left( 1 + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
C)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 40 } + \cdots \right)$ 
D)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } + \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$ 
E)  $y = A \left( 1 - \frac { x ^ { 4 } } { 12 } - \frac { x ^ { 6 } } { 90 } - \cdots \right) + B \left( x + \frac { x ^ { 3 } } { 6 } - \frac { x ^ { 5 } } { 40 } - \cdots \right)$

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

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

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

The general solution of the Bernoulli equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$ is

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

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

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

The orthogonal trajectories of $2 x ^ { 2 } + 3 y ^ { 2 } = c$ are

Which of the following is an exact differential equation?

The order and degree of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - 5 x$ are

The general solution of the Bernoulli equation $3 \left( 1 + x ^ { 2 } \right) y ^ { \prime } + 2 x y = 2 x y ^ { 4 }$ is

The general solution of the exact differential equation $\left( 3 x ^ { 2 } - 6 x y \right) d x + \left( 12 y - 3 x ^ { 2 } \right) d y = 0$ is

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

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

A solution of $y ^ { \prime } + 2 y = 0$ is

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

The order and degree of the differential equation $4 \left( y ^ { \prime \prime } \right) ^ { 2 } = \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right) ^ { 3 }$ are

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

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

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

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

Preparing for Calculus

Limits and Continuity

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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

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Directional Derivatives, Gradients, and Extrema

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Exam 17: Differential Equations

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

The general solution of the first-order differential equation xy′=x2+3yx y ^ { \prime } = x ^ { 2 } + 3 yxy′=x2+3y is

The general solution of the Bernoulli equation y′+2xy+xy4=0y ^ { \prime } + 2 x y + x y ^ { 4 } = 0y′+2xy+xy4=0 is

The general solution of (xeyx+y)dx−xdy=0\left( x e ^ { \frac { y } { x } } + y \right) d x - x d y = 0(xexy​+y)dx−xdy=0 is

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

The general solution of the first-order differential equation x3y′+(2−3x2)y=x3x ^ { 3 } y ^ { \prime } + \left( 2 - 3 x ^ { 2 } \right) y = x ^ { 3 }x3y′+(2−3x2)y=x3 is

The orthogonal trajectories of 2x2+3y2=c2 x ^ { 2 } + 3 y ^ { 2 } = c2x2+3y2=c are

Which of the following is an exact differential equation?

The order and degree of the differential equation d2ydx2=−5x\frac { d ^ { 2 } y } { d x ^ { 2 } } = - 5 xdx2d2y​=−5x are

The general solution of the Bernoulli equation 3(1+x2)y′+2xy=2xy43 \left( 1 + x ^ { 2 } \right) y ^ { \prime } + 2 x y = 2 x y ^ { 4 }3(1+x2)y′+2xy=2xy4 is

The general solution of the exact differential equation (3x2−6xy)dx+(12y−3x2)dy=0\left( 3 x ^ { 2 } - 6 x y \right) d x + \left( 12 y - 3 x ^ { 2 } \right) d y = 0(3x2−6xy)dx+(12y−3x2)dy=0 is

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

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

A solution of y′+2y=0y ^ { \prime } + 2 y = 0y′+2y=0 is

The general solution of the Bernoulli equation (1−x3)y′−2(1+x)y=y32\left( 1 - x ^ { 3 } \right) y ^ { \prime } - 2 ( 1 + x ) y = y ^ { \frac { 3 } { 2 } }(1−x3)y′−2(1+x)y=y23​ is

The order and degree of the differential equation 4(y′′)2=(1+(y′)2)34 \left( y ^ { \prime \prime } \right) ^ { 2 } = \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right) ^ { 3 }4(y′′)2=(1+(y′)2)3 are

The general solution of the exact differential equation (2xy+1y)dx+(y−xy2)dy=0\left( \frac { 2 x y + 1 } { y } \right) d x + \left( \frac { y - x } { y ^ { 2 } } \right) d y = 0(y2xy+1​)dx+(y2y−x​)dy=0 is

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

A solution of y=xy′+1−(y′)2y = x y ^ { \prime } + \sqrt { 1 - \left( y ^ { \prime } \right) ^ { 2 } }y=xy′+1−(y′)2​ is

The general solution of the exact differential equation (2xy+1y)dx+(y−xy2)dy=0\left( \frac { 2 x y + 1 } { y } \right) d x + \left( \frac { y - x } { y ^ { 2 } } \right) d y = 0(y2xy+1​)dx+(y2y−x​)dy=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

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

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

The general solution of the Bernoulli equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$ is

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

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

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

The orthogonal trajectories of $2 x ^ { 2 } + 3 y ^ { 2 } = c$ are

The order and degree of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - 5 x$ are

The general solution of the Bernoulli equation $3 \left( 1 + x ^ { 2 } \right) y ^ { \prime } + 2 x y = 2 x y ^ { 4 }$ is

The general solution of the exact differential equation $\left( 3 x ^ { 2 } - 6 x y \right) d x + \left( 12 y - 3 x ^ { 2 } \right) d y = 0$ is

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

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

A solution of $y ^ { \prime } + 2 y = 0$ is

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

The order and degree of the differential equation $4 \left( y ^ { \prime \prime } \right) ^ { 2 } = \left( 1 + \left( y ^ { \prime } \right) ^ { 2 } \right) ^ { 3 }$ are

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

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

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

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