Question 1

Find the particular solution of $\frac { d y } { d x } = x ^ { 2 }$ ,given y = 18 when x = 1.

Accepted Answer

A)  $y = \frac { x ^ { 3 } } { 3 } - 53$ 
B)  $y = \frac { x ^ { 3 } } { 3 } - \frac { 53 } { 3 }$ 
C)  $y = \frac { x ^ { 3 } } { 3 } + 18$ 
D)  $y = \frac { x ^ { 3 } } { 3 } + \frac { 53 } { 3 }$ 
A)  $y = \frac { x ^ { 3 } } { 3 } - 53$ 
B)  $y = \frac { x ^ { 3 } } { 3 } - \frac { 53 } { 3 }$ 
C)  $y = \frac { x ^ { 3 } } { 3 } + 18$ 
D)  $y = \frac { x ^ { 3 } } { 3 } + \frac { 53 } { 3 }$ D

Question 2

Solve the given first-order linear initial value problem. $- y _ { n + 1 } = y _ { n } ; y _ { 0 } = 1$

Accepted Answer

A)  $y _ { n } = 1$ 
B)  $y _ { n } = ( - 1 ) ^ { n + 1 }$ 
C)  $y _ { n } = ( - 1 ) ^ { n }$ 
D)  $y _ { n } = ( - 1 ) ^ { n } + 1$ 
A)  $y _ { n } = 1$ 
B)  $y _ { n } = ( - 1 ) ^ { n + 1 }$ 
C)  $y _ { n } = ( - 1 ) ^ { n }$ 
D)  $y _ { n } = ( - 1 ) ^ { n } + 1$ C

Question 3

The first five terms of the initial value problem $y _ { n } = y _ { n - 1 } ^ { 2 } ; y _ { 0 } = 2$ are 2,4,8,16,and 32.

Accepted Answer

A) True 
 B)False  True

Question 4

Find constants A and B so that the given expression $y _ { n }$ satisfies the specified difference equation. $n ^ { 2 } y _ { n } + n y _ { n - 1 } = 4 n - 2 n ^ { 3 } ; y _ { n } = A n + B$

Accepted Answer

A)  $A = 2 , B = 2$ 
B)  $A = - 2 , B = - 2$ 
C)  $A = 2 , B = - 2$ 
D)  $A = - 2 , B = 2$ 
A)  $A = 2 , B = 2$ 
B)  $A = - 2 , B = - 2$ 
C)  $A = 2 , B = - 2$ 
D)  $A = - 2 , B = 2$

Question 5

The intensity of light $I ( d )$ at a depth d below the surface of a body of water changes at a rate proportional to I.If the intensity at a depth of $d = 2.4$ feet is half of the surface intensity $I _ { 0 }$ to the nearest tenth of a foot,at what depth is the intensity $15 \%$ of $I _ { 0 } ?$

Accepted Answer

A) 6.6 feet 
B) 9.9 feet 
C) 13.2 feet 
D) 7.9 feet 
A) 6.6 feet 
B) 9.9 feet 
C) 13.2 feet 
D) 7.9 feet

Question 6

Find the general solution of $\frac { d y } { d x } = e ^ { 7 x }$ .

Accepted Answer

A)  $y = \frac { e ^ { 7 x } } { 7 } + C$ 
B)  $y = \frac { ( 7 x ) ^ { e } } { C }$ 
C)  $y = 7 x e ^ { 7 x } + C$ 
D)  $y = \frac { C e ^ { 7 x } } { 7 }$ 
A)  $y = \frac { e ^ { 7 x } } { 7 } + C$ 
B)  $y = \frac { ( 7 x ) ^ { e } } { C }$ 
C)  $y = 7 x e ^ { 7 x } + C$ 
D)  $y = \frac { C e ^ { 7 x } } { 7 }$

Question 7

Write a differential equation describing the given situation.Define all variables you introduce.(Do not try to solve the differential equation at this time.)An investment grows at a rate of 3% of its size.

Accepted Answer

A) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 Q$ 
B) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 t$ 
C) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.3 Q$ 
D) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 3 Q$ 
A) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 Q$ 
B) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 t$ 
C) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.3 Q$ 
D) Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 3 Q$

Question 8

The price of a certain house is currently $260,000.Suppose it is estimated that after t months,the price $$p ( t )$$ will be increasing at the rate of $0.01 p ( t ) + 1,000 t$ dollars per month.
In 7 months from now,to the nearest whole dollar,the price of the house will be $303,934.

Accepted Answer

A) True 
 B)False

Question 9

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 1 )$ of the given initial value problem.Round your answer to two decimal places. $y ^ { \prime } = \frac { x - y } { 4 x + y } ; y ( 0 ) = 4$

Accepted Answer

A) 1.64 
B) 4.92 
C) 6.56 
D) 3.28 
A) 1.64 
B) 4.92 
C) 6.56 
D) 3.28

Question 10

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { y } { 20 x } = \sqrt [ 20 ] { x ^ { 19 } } e ^ { x }$

Accepted Answer

A)  $y = \frac { e ^ { x } } { x ^ { 1 / 20 } } + C$ 
B)  $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 20 } }$ 
C)  $y = x ^ { 1 / 20 } e ^ { x } ( x - 1 ) + C$ 
D)  $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 1 / 20 } }$ 
A)  $y = \frac { e ^ { x } } { x ^ { 1 / 20 } } + C$ 
B)  $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 20 } }$ 
C)  $y = x ^ { 1 / 20 } e ^ { x } ( x - 1 ) + C$ 
D)  $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 1 / 20 } }$

Question 11

Find the general solution of $\frac { d y } { d x } = 7 y$ .

Accepted Answer

A) y = ln x + C 
B)  $y = e ^ { 7 x } + C$ 
C) y = 7x + C 
D)  $y = C e ^ { 7 x }$ 
A) y = ln x + C 
B)  $y = e ^ { 7 x } + C$ 
C) y = 7x + C 
D)  $y = C e ^ { 7 x }$

Question 12

Find the general solution of $\frac { d y } { d x } = \frac { 19 x } { y ^ { 2 } }$ .

Accepted Answer

A)  $y = C$ 
B)  $y = \sqrt [ 3 ] { \frac { 57 x ^ { 2 } } { 2 } + C }$ 
C)  $y = \sqrt [ 3 ] { \frac { ( 38 x ) ^ { 2 } } { 3 } + C }$ 
D)  $y = \frac { 19 x ^ { 2 } } { 2 } + C$ 
A)  $y = C$ 
B)  $y = \sqrt [ 3 ] { \frac { 57 x ^ { 2 } } { 2 } + C }$ 
C)  $y = \sqrt [ 3 ] { \frac { ( 38 x ) ^ { 2 } } { 3 } + C }$ 
D)  $y = \frac { 19 x ^ { 2 } } { 2 } + C$

Question 13

Find an equation for the orthogonal trajectories of the given family of curves. $8 x ^ { 2 } + y = C$

Accepted Answer

A)  $y - \ln x ^ { 16 } = C$ 
B)  $y - x e ^ { 1 / 16 } = C$ 
C)  $y - \ln x ^ { 1 / 6 } = C$ 
D)  $\ln y - x ^ { 1 / 6 } = C$ 
A)  $y - \ln x ^ { 16 } = C$ 
B)  $y - x e ^ { 1 / 16 } = C$ 
C)  $y - \ln x ^ { 1 / 6 } = C$ 
D)  $\ln y - x ^ { 1 / 6 } = C$

Question 14

Find the particular solution of $\frac { d y } { d x } = \frac { x ^ { 2 } y ^ { 3 } } { 5 }$ ,given y = 6 when x = 0.

Accepted Answer

A)  $y = \frac { 2 } { \left( x ^ { 3 } - 6 \right) ^ { 2 } }$ 
B)  $y = 4 x ^ { 6 } + 6$ 
C)  $y = \sqrt { \frac { 180 } { 5 - 24 x ^ { 3 } } }$ 
D)  $y = \frac { x ^ { 3 } } { 3 } + 6$ 
A)  $y = \frac { 2 } { \left( x ^ { 3 } - 6 \right) ^ { 2 } }$ 
B)  $y = 4 x ^ { 6 } + 6$ 
C)  $y = \sqrt { \frac { 180 } { 5 - 24 x ^ { 3 } } }$ 
D)  $y = \frac { x ^ { 3 } } { 3 } + 6$

Question 15

Find the general solution of $\frac { d y } { d x } = x ^ { 3 } + 9$ .

Accepted Answer

A) y = 9x + C 
B)  $y = 4 x ^ { 4 } + 9 x + C$ 
C)  $y = 3 x ^ { 2 } + C$ 
D)  $y = \frac { x ^ { 4 } } { 4 } + 9 x + C$ 
A) y = 9x + C 
B)  $y = 4 x ^ { 4 } + 9 x + C$ 
C)  $y = 3 x ^ { 2 } + C$ 
D)  $y = \frac { x ^ { 4 } } { 4 } + 9 x + C$

Question 16

A dead body is discovered at 8:00 A.M.on Tuesday in a basement where the air temperature is $60 ^ { \circ } \mathrm { F }$ The temperature of the body at the time of discovery is $72 ^ { \circ } \mathrm { F }$ and 20 minutes later,the temperature is $71 ^ { \circ } \mathrm { F }$ The time of death was 6:00 A.M.on Tuesday.

Accepted Answer

A) True 
 B)False

Question 17

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = y ^ { 2 } \sqrt { 6 - x }$ ; y = 1 when x = 6.

Accepted Answer

A)  $y = \frac { 2 ( 6 - x ) ^ { 3 / 2 } - 3 } { 3 }$ 
B)  $y = \frac { 3 } { 2 ( 6 + x ) ^ { 3 / 2 } - 3 }$ 
C)  $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } - 3 }$ 
D)  $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } + 3 }$ 
A)  $y = \frac { 2 ( 6 - x ) ^ { 3 / 2 } - 3 } { 3 }$ 
B)  $y = \frac { 3 } { 2 ( 6 + x ) ^ { 3 / 2 } - 3 }$ 
C)  $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } - 3 }$ 
D)  $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } + 3 }$

Question 18

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$

Accepted Answer

A)  $y = \frac { x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$ 
B)  $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 11 } }$ 
C)  $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$ 
D)  $y = \frac { 11 x ^ { 2 } } { 4 } + \frac { C } { x ^ { 9 } }$ 
A)  $y = \frac { x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$ 
B)  $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 11 } }$ 
C)  $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$ 
D)  $y = \frac { 11 x ^ { 2 } } { 4 } + \frac { C } { x ^ { 9 } }$

Question 19

A desert preserve in the southwest is known for its populations of coyotes and road runners.The growth rate for each population can be modeled by this pair of differential equations: $\begin{array} { l } 
\frac { d R } { d t } = 2 R - 0.2 R C \
\frac { d C } { d t } = - 1.4 C + 0.02 R C
\end{array}$ where C is the number of coyotes and R is the number of road runners.Find the equilibrium populations for this model.(Hint: These are the populations for which $\frac { d R } { d t } = \frac { d C } { d t } = 0$ ).

Accepted Answer

A) 60 coyotes,15 road runners 
B) 10 coyotes,70 road runners 
C) 15 coyotes,60 road runners 
D) 70 coyotes,10 road runners 
A) 60 coyotes,15 road runners 
B) 10 coyotes,70 road runners 
C) 15 coyotes,60 road runners 
D) 70 coyotes,10 road runners

Question 20

Find the particular solution of the given differential equation that satisfies the given condition. $\frac { d y } { d x } - y = 2 x ^ { 2 } ; y = 1 \text { when } x = 3$

Accepted Answer

A)  $y = \frac { 35 e ^ { x - 3 } } { x ^ { 2 } + 2 x + 2 }$ 
B)  $y = 35 e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$ 
C)  $y = 35 e ^ { x } + 2 \left( x ^ { 2 } + 2 x + 2 \right)$ 
D)  $y = C e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$ 
A)  $y = \frac { 35 e ^ { x - 3 } } { x ^ { 2 } + 2 x + 2 }$ 
B)  $y = 35 e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$ 
C)  $y = 35 e ^ { x } + 2 \left( x ^ { 2 } + 2 x + 2 \right)$ 
D)  $y = C e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$

Find the particular solution of $\frac { d y } { d x } = x ^ { 2 }$ ,given y = 18 when x = 1.

Solve the given first-order linear initial value problem. $- y _ { n + 1 } = y _ { n } ; y _ { 0 } = 1$

The first five terms of the initial value problem $y _ { n } = y _ { n - 1 } ^ { 2 } ; y _ { 0 } = 2$ are 2,4,8,16,and 32.

Find constants A and B so that the given expression $y _ { n }$ satisfies the specified difference equation. $n ^ { 2 } y _ { n } + n y _ { n - 1 } = 4 n - 2 n ^ { 3 } ; y _ { n } = A n + B$

The intensity of light $I ( d )$ at a depth d below the surface of a body of water changes at a rate proportional to I.If the intensity at a depth of $d = 2.4$ feet is half of the surface intensity $I _ { 0 }$ to the nearest tenth of a foot,at what depth is the intensity $15 \%$ of $I _ { 0 } ?$

Find the general solution of $\frac { d y } { d x } = e ^ { 7 x }$ .

Write a differential equation describing the given situation.Define all variables you introduce.(Do not try to solve the differential equation at this time.)An investment grows at a rate of 3% of its size.

The price of a certain house is currently $260,000.Suppose it is estimated that after t months,the price $p ( t )$ will be increasing at the rate of $0.01 p ( t ) + 1,000 t$ dollars per month. In 7 months from now,to the nearest whole dollar,the price of the house will be $303,934.

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 1 )$ of the given initial value problem.Round your answer to two decimal places. $y ^ { \prime } = \frac { x - y } { 4 x + y } ; y ( 0 ) = 4$

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { y } { 20 x } = \sqrt [ 20 ] { x ^ { 19 } } e ^ { x }$

Find the general solution of $\frac { d y } { d x } = 7 y$ .

Find the general solution of $\frac { d y } { d x } = \frac { 19 x } { y ^ { 2 } }$ .

Find an equation for the orthogonal trajectories of the given family of curves. $8 x ^ { 2 } + y = C$

Find the particular solution of $\frac { d y } { d x } = \frac { x ^ { 2 } y ^ { 3 } } { 5 }$ ,given y = 6 when x = 0.

Find the general solution of $\frac { d y } { d x } = x ^ { 3 } + 9$ .

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = y ^ { 2 } \sqrt { 6 - x }$ ; y = 1 when x = 6.

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$

Find the particular solution of the given differential equation that satisfies the given condition. $\frac { d y } { d x } - y = 2 x ^ { 2 } ; y = 1 \text { when } x = 3$

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Exam 8: Differential Equations

Find the particular solution of dydx=x2\frac { d y } { d x } = x ^ { 2 }dxdy​=x2 ,given y = 18 when x = 1.

Solve the given first-order linear initial value problem. −yn+1=yn;y0=1- y _ { n + 1 } = y _ { n } ; y _ { 0 } = 1−yn+1​=yn​;y0​=1

The first five terms of the initial value problem yn=yn−12;y0=2y _ { n } = y _ { n - 1 } ^ { 2 } ; y _ { 0 } = 2yn​=yn−12​;y0​=2 are 2,4,8,16,and 32.

Find constants A and B so that the given expression yny _ { n }yn​ satisfies the specified difference equation. n2yn+nyn−1=4n−2n3;yn=An+Bn ^ { 2 } y _ { n } + n y _ { n - 1 } = 4 n - 2 n ^ { 3 } ; y _ { n } = A n + Bn2yn​+nyn−1​=4n−2n3;yn​=An+B

Find the general solution of dydx=e7x\frac { d y } { d x } = e ^ { 7 x }dxdy​=e7x .

Write a differential equation describing the given situation.Define all variables you introduce.(Do not try to solve the differential equation at this time.)An investment grows at a rate of 3% of its size.

Use Euler's method with the step size h=0.2h = 0.2h=0.2 to estimate the solution y(1)y ( 1 )y(1) of the given initial value problem.Round your answer to two decimal places. y′=x−y4x+y;y(0)=4y ^ { \prime } = \frac { x - y } { 4 x + y } ; y ( 0 ) = 4y′=4x+yx−y​;y(0)=4

Find the general solution of the given first-order linear differential equation. dydx+y20x=x1920ex\frac { d y } { d x } + \frac { y } { 20 x } = \sqrt [ 20 ] { x ^ { 19 } } e ^ { x }dxdy​+20xy​=20x19​ex

Find the general solution of dydx=7y\frac { d y } { d x } = 7 ydxdy​=7y .

Find the general solution of dydx=19xy2\frac { d y } { d x } = \frac { 19 x } { y ^ { 2 } }dxdy​=y219x​ .

Find an equation for the orthogonal trajectories of the given family of curves. 8x2+y=C8 x ^ { 2 } + y = C8x2+y=C

Find the particular solution of dydx=x2y35\frac { d y } { d x } = \frac { x ^ { 2 } y ^ { 3 } } { 5 }dxdy​=5x2y3​ ,given y = 6 when x = 0.

Find the general solution of dydx=x3+9\frac { d y } { d x } = x ^ { 3 } + 9dxdy​=x3+9 .

Find the particular solution of the given differential equation that satisfies the indicated condition: dydx=y26−x\frac { d y } { d x } = y ^ { 2 } \sqrt { 6 - x }dxdy​=y26−x​ ; y = 1 when x = 6.

Find the general solution of the given first-order linear differential equation. dydx+9yx=4x\frac { d y } { d x } + \frac { 9 y } { x } = 4 xdxdy​+x9y​=4x

Find the particular solution of the given differential equation that satisfies the given condition. dydx−y=2x2;y=1 when x=3\frac { d y } { d x } - y = 2 x ^ { 2 } ; y = 1 \text { when } x = 3dxdy​−y=2x2;y=1 when x=3

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Find the particular solution of $\frac { d y } { d x } = x ^ { 2 }$ ,given y = 18 when x = 1.

Solve the given first-order linear initial value problem. $- y _ { n + 1 } = y _ { n } ; y _ { 0 } = 1$

The first five terms of the initial value problem $y _ { n } = y _ { n - 1 } ^ { 2 } ; y _ { 0 } = 2$ are 2,4,8,16,and 32.

Find constants A and B so that the given expression $y _ { n }$ satisfies the specified difference equation. $n ^ { 2 } y _ { n } + n y _ { n - 1 } = 4 n - 2 n ^ { 3 } ; y _ { n } = A n + B$

Find the general solution of $\frac { d y } { d x } = e ^ { 7 x }$ .

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 1 )$ of the given initial value problem.Round your answer to two decimal places. $y ^ { \prime } = \frac { x - y } { 4 x + y } ; y ( 0 ) = 4$

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { y } { 20 x } = \sqrt [ 20 ] { x ^ { 19 } } e ^ { x }$

Find the general solution of $\frac { d y } { d x } = 7 y$ .

Find the general solution of $\frac { d y } { d x } = \frac { 19 x } { y ^ { 2 } }$ .

Find an equation for the orthogonal trajectories of the given family of curves. $8 x ^ { 2 } + y = C$

Find the particular solution of $\frac { d y } { d x } = \frac { x ^ { 2 } y ^ { 3 } } { 5 }$ ,given y = 6 when x = 0.

Find the general solution of $\frac { d y } { d x } = x ^ { 3 } + 9$ .

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = y ^ { 2 } \sqrt { 6 - x }$ ; y = 1 when x = 6.

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$

Find the particular solution of the given differential equation that satisfies the given condition. $\frac { d y } { d x } - y = 2 x ^ { 2 } ; y = 1 \text { when } x = 3$