Exam 10: Differential Equations

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $( 3 t + 7 ) y ^ { \prime } + 3 y = 4 t$ . Is this derivative correct: $\frac { d } { d t } [ ( 3 t + 7 ) y ] = 4 t$ ?

(True/False)

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Question 81

Given the differential equation with the given initial condition: $y y ^ { \prime } = \operatorname { te } ^ { \mathrm { t } ^ { 2 } } ; \quad y ( 0 ) = 1$ is this the solution $y = e ^ { t ^ { 2 } / 2 } ?$

(True/False)

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Question 82

Depending on the type of soil there is a constant M that represents the maximum amount of water the soil can absorb per cubic ft. If the rate of absorption is proportional to the difference between the maximum amount of water that could be absorbed and the amount of water that has been absorbed, write a differential equation satisfied by y = f(t), the amount of water in the soil at time t. Does this equation accurately describe this situation: $\mathrm { y } ^ { \prime } = \mathrm { k } ( \mathrm { M } - \mathrm { y } ) ; \mathrm { k } > 0 ?$

(True/False)

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Question 83

Given $y ^ { \prime }$ = y - 3. On a ty-coordinate system sketch the solutions corresponding to the initial conditions $y ( 0 ) = 0$ and $y ( 0 ) = 4$ Does this graph represent the situation? $Given y ^ { \prime } = y - 3. On a ty-coordinate system sketch the solutions corresponding to the initial conditions y ( 0 ) = 0 and y ( 0 ) = 4 Does this graph represent the situation?$

(True/False)

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Question 84

Given the differential equation: $y ^ { \prime } = e ^ { 2 t } - 1$ , is this the solution $y = \frac { e ^ { 2 t } } { 2 } - t + C ?$

(True/False)

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Question 85

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 y ; \quad y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales ?

(Multiple Choice)

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Question 86

Which of the following functions solves the differential equation: $y ^ { \prime } = y ^ { 2 } ?$

(Multiple Choice)

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Question 87

Solve the equation using an integrating factor. - $\mathrm { y } ^ { \prime } + \frac { 4 } { \mathrm { t } } \mathrm { y } = \mathrm { t } ^ { 4 }$ , t > 0

(Multiple Choice)

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Question 88

Consider the differential equation $y ^ { \prime }$ = $t ^ { 3 }$ (y + 3). Which of the following statements is/are true? (I) f(t) = -3 is a constant solution to this differential equation. (II) f(t) = 0 is a constant solution to this differential equation. (III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then $f ^ { \prime }$ (1) = 3.

(Multiple Choice)

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Question 89

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 ( 10 - y ) ; y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales.

(Multiple Choice)

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Question 90

Find a constant solution of $y ^ { \prime } = t ( y - 1 )$ Enter just an integer.

(Short Answer)

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Question 91

One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. $y ^ { \prime } = y ^ { 2 } - 4 ; y ( 0 ) = - 3 ; y ( 0 ) = - 1 ; y ( 0 ) = 1 ; y ( 0 ) = 3$ Do these graphs represent the situation? $One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y ^ { \prime } = y ^ { 2 } - 4 ; y ( 0 ) = - 3 ; y ( 0 ) = - 1 ; y ( 0 ) = 1 ; y ( 0 ) = 3 Do these graphs represent the situation?$

(True/False)

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Question 92

Suppose water is seeping from an underground storage facility at a rate that is proportional to the square amount of water present. If f(t) = y is the amount of water present at time t, find a differential equation describing the situation.

(Multiple Choice)

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Question 93

A tank contains 2000 L of a solution consisting of 50 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 10L/s, and the mixture (kept uniform by stirring) is pumped out at the same rate. How long will it be until only 5 kg of salt remain in the tank?

(Multiple Choice)

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Question 94

A fly population increases at a rate proportional to the amount present. After two years the population has doubled. After three years it is 20,000. Find the number of flies initially present. Enter just an integer.

(Short Answer)

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Question 95

Use Euler's method with n = 5 to approximate the solution f(t) to $y ^ { \prime }$ = 5 - y, $y ( 0 ) = 1$ for $0 \leq t \leq 1$ . Estimate f(1). Enter just a real number rounded off to two decimal places.

(Short Answer)

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Question 96

There is a differential equation that is a mathematical model of the situation in which the time rate of change in the population of a certain organism is proportional to the product of the current population and the difference between the current population and the limiting factor of 100,000. Is this the equation $\frac { \mathrm { dP } } { \mathrm { dt } } = \mathrm { kP } ( 100,000 - \mathrm { P } ) ?$

(True/False)

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Question 97

Combine the terms y and $y ^ { \prime }$ into the derivative of a product, then solve the equation. $\frac { y ^ { \prime } } { t ^ { 3 } } - \frac { 3 y } { t ^ { 4 } } = t$ Is this the solution: $y = \frac { t ^ { 5 } } { 2 } + C ^ { 3 } ?$

(True/False)

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Question 98

A certain developing country has a population of 500,000. The yearly rate of increase of literacy among the people is proportional to the number of illiterate people in the population. Letting f(t) represent the number of literate people, determine the differential equation that f(t) satisfies. (Let k represent a positive constant.)

(Multiple Choice)

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Question 99

Which of the following functions solves the differential equation: $y ^ { \prime } = - 6 x y$ ?

(Multiple Choice)

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Question 100

Showing 81 - 100 of 126

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $( 3 t + 7 ) y ^ { \prime } + 3 y = 4 t$ . Is this derivative correct: $\frac { d } { d t } [ ( 3 t + 7 ) y ] = 4 t$ ?

Given the differential equation with the given initial condition: $y y ^ { \prime } = \operatorname { te } ^ { \mathrm { t } ^ { 2 } } ; \quad y ( 0 ) = 1$ is this the solution $y = e ^ { t ^ { 2 } / 2 } ?$

Given the differential equation: $y ^ { \prime } = e ^ { 2 t } - 1$ , is this the solution $y = \frac { e ^ { 2 t } } { 2 } - t + C ?$

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 y ; \quad y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales ?

Which of the following functions solves the differential equation: $y ^ { \prime } = y ^ { 2 } ?$

Solve the equation using an integrating factor. - $\mathrm { y } ^ { \prime } + \frac { 4 } { \mathrm { t } } \mathrm { y } = \mathrm { t } ^ { 4 }$ , t > 0

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 ( 10 - y ) ; y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales.

Find a constant solution of $y ^ { \prime } = t ( y - 1 )$ Enter just an integer.

Suppose water is seeping from an underground storage facility at a rate that is proportional to the square amount of water present. If f(t) = y is the amount of water present at time t, find a differential equation describing the situation.

A tank contains 2000 L of a solution consisting of 50 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 10L/s, and the mixture (kept uniform by stirring) is pumped out at the same rate. How long will it be until only 5 kg of salt remain in the tank?

A fly population increases at a rate proportional to the amount present. After two years the population has doubled. After three years it is 20,000. Find the number of flies initially present. Enter just an integer.

Use Euler's method with n = 5 to approximate the solution f(t) to $y ^ { \prime }$ = 5 - y, $y ( 0 ) = 1$ for $0 \leq t \leq 1$ . Estimate f(1). Enter just a real number rounded off to two decimal places.

Combine the terms y and $y ^ { \prime }$ into the derivative of a product, then solve the equation. $\frac { y ^ { \prime } } { t ^ { 3 } } - \frac { 3 y } { t ^ { 4 } } = t$ Is this the solution: $y = \frac { t ^ { 5 } } { 2 } + C ^ { 3 } ?$

Which of the following functions solves the differential equation: $y ^ { \prime } = - 6 x y$ ?

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 10: Differential Equations

Combine the terms y and y′y ^ { \prime }y′ into the derivative of a product: (3t+7)y′+3y=4t( 3 t + 7 ) y ^ { \prime } + 3 y = 4 t(3t+7)y′+3y=4t . Is this derivative correct: ddt[(3t+7)y]=4t\frac { d } { d t } [ ( 3 t + 7 ) y ] = 4 tdtd​[(3t+7)y]=4t ?

Given the differential equation with the given initial condition: yy′=te⁡t2;y(0)=1y y ^ { \prime } = \operatorname { te } ^ { \mathrm { t } ^ { 2 } } ; \quad y ( 0 ) = 1yy′=tet2;y(0)=1 is this the solution y=et2/2?y = e ^ { t ^ { 2 } / 2 } ?y=et2/2?

Given y′y ^ { \prime }y′ = y - 3. On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0)=0y ( 0 ) = 0y(0)=0 and y(0)=4y ( 0 ) = 4y(0)=4 Does this graph represent the situation?

Given the differential equation: y′=e2t−1y ^ { \prime } = e ^ { 2 t } - 1y′=e2t−1 , is this the solution y=e2t2−t+C?y = \frac { e ^ { 2 t } } { 2 } - t + C ?y=2e2t​−t+C?

The annual sales y (in millions of dollars) of a company satisfy the differential equation dydt=0.2y;y(0)=2\frac { d y } { d t } = 0.2 y ; \quad y ( 0 ) = 2dtdy​=0.2y;y(0)=2 . Which of the following is a verbal description of the rate of change of annual sales ?

Which of the following functions solves the differential equation: y′=y2?y ^ { \prime } = y ^ { 2 } ?y′=y2?

Solve the equation using an integrating factor. - y′+4ty=t4\mathrm { y } ^ { \prime } + \frac { 4 } { \mathrm { t } } \mathrm { y } = \mathrm { t } ^ { 4 }y′+t4​y=t4 , t > 0

The annual sales y (in millions of dollars) of a company satisfy the differential equation dydt=0.2(10−y);y(0)=2\frac { d y } { d t } = 0.2 ( 10 - y ) ; y ( 0 ) = 2dtdy​=0.2(10−y);y(0)=2 . Which of the following is a verbal description of the rate of change of annual sales.

Find a constant solution of y′=t(y−1)y ^ { \prime } = t ( y - 1 )y′=t(y−1) Enter just an integer.

Suppose water is seeping from an underground storage facility at a rate that is proportional to the square amount of water present. If f(t) = y is the amount of water present at time t, find a differential equation describing the situation.

A tank contains 2000 L of a solution consisting of 50 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 10L/s, and the mixture (kept uniform by stirring) is pumped out at the same rate. How long will it be until only 5 kg of salt remain in the tank?

A fly population increases at a rate proportional to the amount present. After two years the population has doubled. After three years it is 20,000. Find the number of flies initially present. Enter just an integer.

Use Euler's method with n = 5 to approximate the solution f(t) to y′y ^ { \prime }y′ = 5 - y, y(0)=1y ( 0 ) = 1y(0)=1 for 0≤t≤10 \leq t \leq 10≤t≤1 . Estimate f(1). Enter just a real number rounded off to two decimal places.

Which of the following functions solves the differential equation: y′=−6xyy ^ { \prime } = - 6 x yy′=−6xy ?

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $( 3 t + 7 ) y ^ { \prime } + 3 y = 4 t$ . Is this derivative correct: $\frac { d } { d t } [ ( 3 t + 7 ) y ] = 4 t$ ?

Given the differential equation with the given initial condition: $y y ^ { \prime } = \operatorname { te } ^ { \mathrm { t } ^ { 2 } } ; \quad y ( 0 ) = 1$ is this the solution $y = e ^ { t ^ { 2 } / 2 } ?$

Given the differential equation: $y ^ { \prime } = e ^ { 2 t } - 1$ , is this the solution $y = \frac { e ^ { 2 t } } { 2 } - t + C ?$

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 y ; \quad y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales ?

Which of the following functions solves the differential equation: $y ^ { \prime } = y ^ { 2 } ?$

Solve the equation using an integrating factor. - $\mathrm { y } ^ { \prime } + \frac { 4 } { \mathrm { t } } \mathrm { y } = \mathrm { t } ^ { 4 }$ , t > 0

The annual sales y (in millions of dollars) of a company satisfy the differential equation $\frac { d y } { d t } = 0.2 ( 10 - y ) ; y ( 0 ) = 2$ . Which of the following is a verbal description of the rate of change of annual sales.

Find a constant solution of $y ^ { \prime } = t ( y - 1 )$ Enter just an integer.

Use Euler's method with n = 5 to approximate the solution f(t) to $y ^ { \prime }$ = 5 - y, $y ( 0 ) = 1$ for $0 \leq t \leq 1$ . Estimate f(1). Enter just a real number rounded off to two decimal places.

Which of the following functions solves the differential equation: $y ^ { \prime } = - 6 x y$ ?