Deck 11: Differential Equations

Question 1

Which of the following functions are solutions to the differential equation $\frac { d y } { d x } = - \frac { 1 } { 6 } y$ ? (Mark all correct answers.)

A) $y = \sin ( 6 x ) + \cos ( 6 x )$
B) $y = 6 \cos x - \sin x$
C) $y = e ^ { x / 6 }$
D) $y = e ^ { - x / 6 }$

Answer 1

$y = e ^ { - x / 6 }$

Question 2

Select the value(s)of w for which $y = e ^ { w t }$ satisfies $\frac { d ^ { 2 } y } { d t ^ { 2 } } - 49 y = 0$ .

A)7
B)-7
C)49
D)-49
E)0

Answer 2

7
-7

Question 3

One of the solutions to the differential equation $\frac { d y } { d x } = x - \frac { 1 } { 5 } y$ is a straight line.Take a point $\left( x _ { 0 } , y _ { 0 } \right)$ not on the line.Can a solution curve through $\left( x _ { 0 } , y _ { 0 } \right)$ cross the line?

Answer 3

False

Question 4

Suppose $y ^ { \prime } = 2 y - x$ and $y ( 0 ) = 2$ .Estimate $y ( 0.05 )$ and $y ( 0.1 )$ using the slope given by the differential equation.

A)4 and 3.9
B)5.95 and 17.75
C)3.95 and 3.9
D)7.95 and 17.95
E)2.2 and 2.4175

Question 5

For what value(s)of n (if any)is $y = e ^ { n x }$ a solution to the differential equation $- \frac { 1 } { 2 } y ^ { \prime \prime } + y ^ { \prime } + 4 y = 0$ ?

A)2
B)-2
C)4
D)-4
E)0

Question 6

Mark all the true statements about the differential equation: $\frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } + B y = 0$

A)It is a first order differential equation.
B)It is a second order differential equation.
C)Any solutions to the differential equation would be of the form x = a.
D)Any solutions to the differential equation would be of the form y = f(x).

Question 7

Which of the following functions are solutions to the differential equation $\frac { d y } { d x } = \frac { y } { 3 }$ ? (Mark all correct answers.)

A) $y = \sin ( 3 x ) + \cos ( 3 x )$
B) $y = 3 \cos x - \sin x$
C) $y = e ^ { x / 3 }$
D) $y = e ^ { - x / 3 }$

Question 8

Suppose $y ^ { \prime } = y$ and $y ( 1 ) = 4$ .What is the best approximation of $y ( 1.2 )$ .

A)0.8
B)3.6
C)4.8
D)16

Question 9

Suppose that the function P(t)satisfies the differential equation $P ^ { \prime } ( t ) = P ( t ) ( 4 - P ( t ) )$ with the initial condition P(0)= 1.Consider the behavior of the graph of $P ( t )$ near a point $t _ { 0 }$ , where $P \left( t _ { 0 } \right) = 4$ (if such a point exists).Is the following graph consistent with P(t)?

Question 10

Suppose that the function P(t)satisfies the differential equation $P ^ { \prime } ( t ) = P ( t ) ( 3 - P ( t ) )$ with the initial condition P(0)= 1.Find P"(0).

Question 11

Suppose that the function P(t)satisfies the differential equation with the initial condition P(0)= 4.Which of the following is a possible graph for P(t)for small t > 0?

Question 12

Given $\frac { d ^ { 2 } s } { d t ^ { 2 } } = - 9.8$ , find S if the initial velocity is 10 m/sec upward and the initial position is 5 m above the ground.

A) $S = - 4.9 t ^ { 2 } + 10 t + 5$
B) $S = 4.9 t ^ { 2 } + 10 t + 5$
C) $S = - 9.8 t ^ { 2 } + 10 t + 5$
D) $S = 9.8 t ^ { 2 } + 10 t + 5$

Question 13

Does $y = 4 \cos ( 4 t )$ satisfy $\frac { d ^ { 2 } y } { d t ^ { 2 } } + 4 y = 0$ ?

Question 14

Which of the following equations corresponds with the slope field shown below?
I. II. III. IV.None of them

Question 15

Which of the following equations corresponds with the slope field shown below?

A) $y ^ { \prime } = x y + 1$
B) $y ^ { \prime } = \sin x$
C) $y ^ { \prime } = x e ^ { - y }$
D) $y ^ { \prime } = y ^ { 2 } + 1$
E) $y ^ { \prime } = \sin y$
F) None of them

Question 16

Is $y = e ^ { x } \sin x$ a solution to $\frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } + y = 0$ ?

Question 17

Which of the following equations goes with is on the graph of each of the equations).

A) $y ^ { 2 } - 2 \cos x = 2$
B) $x \sin y + y = 2$
C) $\ln \left| \frac { y } { 1 - y } \right| = 0.71 x + \ln 2$

Question 18

Which of the following functions are solutions to the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - y$ ? (Mark all correct answers)

A) $y = \sin ( 8 x ) + \cos ( 8 x )$
B) $y = 8 \cos x - \sin x$
C) $y = e ^ { x / 8 }$
D) $y = e ^ { - x / 8 }$

Question 19

The slope field for the differential equation $\frac { d y } { d x } = x - y$ is shown below.Consider the solution curve to the differential equation starting at x = 0, y = 5, and ending at x = 5 and approximate the value of y when x is 5.

A)1.25
B)2.5
C)3.75
D)5

Question 20

Which of the following functions are solutions to the differential equation $\frac { d y } { d x } = - 25 y$ ? (Mark all correct answers)

A) $y = \sin ( 5 x ) + \cos ( 5 x )$
B) $y = 5 \cos x - \sin x$
C) $y = e ^ { x / 5 }$
D) $y = e ^ { - x / 5 }$

Question 21

Consider the solution with y(0)= 0 to the differential equation .Use Euler's method with 2 steps to approximate the value of .Give your answer to 1 decimal place.

Question 22

Consider the differential equation Solve the differential equation analytically given that .Then approximate using Euler's method with three steps.Write a few sentences describing the error in Euler's method in this case, and what could be done to decrease the error.

Question 23

Consider the solution with y(0)= 0 to the differential equation $\frac{d y}{d x}=\frac{7}{1+x^{2}}$ .If you use Euler's method with 1 million steps to approximate the exact value of $y ( 1 )$ , will your approximation be an over- or underestimate?

A)An overestimate
B)An underestimate

Question 24

If a slope field for $\frac { d y } { d x }$ has constant slopes where y is constant, what do you know about $\frac { d y } { d x }$ ?

A) $\frac { d y } { d x }$ depends on y only.
B) $\frac { d y } { d x }$ depends on x only.
C) $\frac { d y } { d x }$ must be linear in y.
D) $\frac { d y } { d x }$ must be linear in x.

Question 25

The slope field for the differential equation is shown below.Use Euler's method with N = 3 subdivisions of the interval to approximate the value of y when t = 1, if you start at y = 1 and t = 0.Round your answer to 2 decimal places.

Question 26

Consider the differential equation .Use Euler's method with two subdivisions to approximate the value of y when x = 2 on the solution curve that passes through (1,4).

Question 27

Consider the solution with y(0)= 0 to the differential equation .Compute the exact value of y(1)and then round to 4 decimal places.

Question 28

The slope field for the differential equation $\frac { d y } { d x } = y ( 2 - y )$ is shown below.Use the slope field to estimate the solution starting from y = 3, x = 0, and use it to estimate the value of y when x = 1.

A)1.2
B)1.5
C)1.8
D)2.1

Question 29

Is the equation , for C a constant, a solution to the differential equation ?

Question 30

The slope field for the differential equation is shown below.Starting from the point , use Euler's method with N = 3 subdivisions to approximate the value of y when x = 1.Round to 2 decimal places.

Question 31

Solve the differential equation $\frac { d Q } { d t } = 200 - 0.2 Q$ subject to $Q ( 0 ) = 1500$ .

A) $Q ( t ) = 500 e ^ { - 0.2 t } + 1000$
B) $Q ( t ) = - 500 e ^ { - 0.2 t } + 1000$
C) $Q ( t ) = 1300 e ^ { - 0.2 t } + 200$
D) $Q ( t ) = - 1300 e ^ { - 0.2 t } + 200$

Question 32

The slope field for the differential equation is shown below.Use the Euler's method with N = 4 subdivisions of the interval to approximate the value of y when x = 1, if you start at y = 1 and x = 0.Round your answer to 2 decimal places.

Question 33

On the slope field for the differential equation , sketch the solution curve in the fourth quadrant that goes through the point (0, -1).

Question 34

Use Euler's method with five steps to approximate a solution to the differential equation
. Give both a numerical solution by filling in the table of values, and a graphical solution by plotting your points.

Question 35

Sketch a slope field for the differential equation using the points indicated on the axes.

Question 36

If all the solution curves for $\frac { d y } { d t }$ have $y = - 5$ as a horizontal asymptote, does it follow that either $\lim _ { t \rightarrow \infty } y = 5$ or $\lim _ { t \rightarrow - \infty } y = 5$ ?

Question 37

The slope field for the differential equation $\frac { d y } { d x } = - \frac { x } { y }$ is shown below.If you start from the point $( 0 , - 2 )$ and use Euler's method with N = 3 subdivisions to approximate the value of y when x = 1, is your answer an underestimate or overestimate?

A)An overestimate
B)An underestimate

Question 38

The slope field for the differential equation $\frac { d y } { d t } = 0.5 ( 2 - y )$ is shown below.Consider the solution starting from y = 0.5, t = 0, and use it to estimate the value of y when t = 1.

A)0.5
B)0.8
C)1.1
D)1.4

Question 39

Consider the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$ .If you use Euler's method to approximate the value of y when x = 2 on the solution curve that passes through (1,4), is your approximation an underestimate or overestimate?

A)An underestimate
B)An overestimate

Question 40

For any constant C, $y = 4 x - 16 + C e ^ { - x / 4 }$ is a solution to the differential equation $\frac { d y } { d x } = x - \frac { 1 } { 4 } y$ .Find the solution through the point (0, -4).

A) $y = 4 x - 16 + 16 e ^ { - x / 4 }$
B) $y = 4 x - 16 - 4 e ^ { - x / 4 }$
C) $y = 4 x - 16 + 12 e ^ { - x / 4 }$
D) $y = 4 x - 16 - 16 e ^ { - x / 4 }$

Question 41

Solve $\frac { d y } { d x } = \frac { x } { y }$ if y = 0 when x = 2.

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

Question 42

Find the solution to the differential equation $y ^ { \prime } = \frac { 8 } { 1 + y }$ satisfying $y ( 0 ) = 6$ .Select all that apply.

A) $y + y ^ { 2 } = 8 x + 42$
B) $y + \frac { y ^ { 2 } } { 2 } = 8 x + 24$
C) $\frac { y } { 2 } + \frac { y ^ { 2 } } { 2 } = 8 x + 21$
D) $\frac { ( 1 + y ) ^ { 2 } } { 2 } = 8 x + \frac { 49 } { 2 }$

Question 43

Is $P = 250 + C e ^ { 0.2 t }$ a solution to the differential equation $\frac { d P } { d t } + 0.2 P = 50$ (for some constant C)?

Question 44

Find the solution to the differential equation $x y ^ { \prime } - ( 6 + x ) y = 0$ with $y ( 1 ) = 1$ , $y ( x ) \geq 0$ for all x.

A) $y = x ^ { 7 } e ^ { x }$
B) $y = x ^ { 7 } e ^ { x - 1 }$
C) $y = x ^ { 6 } e ^ { x - 1 }$
D) $y = x ^ { 6 } e ^ { x } + 1 - e$

Question 45

Solve the differential equation subject to and sketch your solution.There is a horizontal asymptote at Q = _____.

Question 46

Solve $\frac { d N } { d t } = 2 - 0.2 \mathrm {~N}$ , with N(0)= 0 and sketch the solution for t $\ge$ 0.There is a horizontal asymptote at N = _____.

Question 47

Find the solution to the differential equation $\frac { d y } { d x } = x y$ satisfying $y ( 0 ) = 3$ .

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

Question 48

Is the slope field for the differential equation $\frac { d y } { d x } = \frac { 4 x ^ { 2 } } { y }$ symmetric about the y-axis?

Question 49

Solve $\frac { d N } { d t } = 4 - 0.2 N$ , with N(0)= 0.

A) $N = 20 e ^ { - 0.2 t }$
B) $N = 4 \left( 1 - e ^ { - 0.2 t } \right)$
C) $N = 4 \left( e ^ { - 0.2 t } - 1 \right)$
D) $N = 20 \left( 1 - e ^ { - 0.2 t } \right)$

Question 50

Find the solution to the differential equation $\frac { d y } { d x } = x \sec y$ if $y = \frac { \pi } { 6 }$ when x = 1.

A) $2 y = \arccos \left( \frac { x ^ { 2 } } { 2 } \right)$
B) $\frac { 3 } { 2 } y = \arctan ( x )$
C) $y - \frac { \pi } { 6 } = \arccos ( x )$
D) $y = \arcsin \left( \frac { x ^ { 2 } } { 2 } \right)$

Question 51

Is $y = \sqrt { C - 2 \cos t }$ a solution to the differential equation $\frac { d y } { d t } = \frac { \sin t } { y ^ { 2 } }$ (for some constant C)?

Question 52

Is $y = - \ln \left( e ^ { - x } - \frac { 15 } { 4 } \right)$ the solution to the differential equation $\frac { d y } { d x } = e ^ { y - x }$ with $y ( \ln 4 ) = - \ln 4$ ?

Question 53

Find the solution to the differential equation $\frac { d y } { d x } = \frac { 6 x } { y }$ passing through $( 1,6 )$ .

A) $y ^ { 2 } = 6 x ^ { 2 } + 30$
B) $6 y ^ { 2 } = x ^ { 2 } + 215$
C) $y ^ { 2 } + 6 x ^ { 2 } = 42$
D) $6 y ^ { 2 } + x ^ { 2 } = 217$

Question 54

Find the solution to the differential equation $\frac { d y } { d x } = \frac { \cos ^ { 2 } y } { x }$ with $y ( 1 ) = - \frac { \pi } { 4 }$ .

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

Question 55

Is $H = C e ^ { t ^ { 2 } } / 2 - 4 t$ the general solution to the differential equation $\frac { d H } { d t } = - 2 H + t H$ ?

Question 56

Solve $\frac { d y } { d t } = y ^ { 2 } + 1$ if y = 1 when t = 0.

A) $y = \arctan \left( t + \frac { \pi } { 4 } \right)$
B) $y = \arctan \left( t - \frac { \pi } { 4 } \right)$
C) $y = \tan \left( t + \frac { \pi } { 4 } \right)$
D) $y = \tan \left( t - \frac { \pi } { 4 } \right)$

Question 57

Solve $\frac { d y } { d x } = \sqrt { 36 - y ^ { 2 } }$ if $y = 3$ when $x = \frac { \pi } { 6 }$ .

A) $y = 6 \sin x$
B) $y = 6 \cos x$
C) $y = 36 \sin x$
D) $y = 36 \cos x$

Question 58

What is the minimum value of the solution to the differential equation $\frac { d y } { d x } = y \cos x$ with $y ( 0 ) = 6$ ?

A)6
B)0
C) $6$ /e
D) $6$ e

Question 59

Solve $\frac { d y } { d x } = \frac { 1 } { \sqrt { x y } }$ if y = 4 when x = 0.

A) $y = \sqrt { 3 \sqrt { x } + 16 }$
B) $y = ( 3 \sqrt { x } + 64 ) ^ { 1 / 3 }$
C) $y = 3 \sqrt { x } + 4$
D) $y = ( 3 \sqrt { x } + 8 ) ^ { 2 / 3 }$

Question 60

Is $y = \frac { a } { b } + C e ^ { - b t }$ the general solution to the differential equation $\frac { d y } { d t } = a - b y$ ?

Question 61

Cesium 137 (Cs¹³⁷)is a short-lived radioactive isotope.It decays at a rate proportional to the amount of itself present and has a half-life of 30 years (i.e., the amount of Cs¹³⁷ remaining t years after A₀ mg of the radioactive isotope is released is given by ).As a result of its operations, a nuclear power plant releases Cs¹³⁷ at a rate of 0.1 mg per year.The plant began its operations in 1990, which we will designate as t = 0.Assume there is no other source of this particular isotope.In the long run, approximately how many mg of Cs¹³⁷ will there be? Round to 2 decimal places.

Question 62

The population of aphids on a rose plant increases at a rate proportional to the number present.In 3 days the population grew from 700 to 1400.How many aphids were there on the day before there were 700 aphids? Round to the nearest whole number.

Question 63

Consider the Hakosalo residence in Oulu, Finland.Assume that heat is lost from the house only through windows and the rate of change of temperature in °F/hr is proportional to the difference in temperature between the outside and the inside.The constant of proportionality is .Assume that it is 10°F outside constantly.On a Thursday at noon the temperature inside the house was 65°F and the heat was turned off until 5 pm.At 5 pm the heat is turned on.The heater generates an amount of energy that would raise the inside temperature by 2°F per hour if there were no heat loss.If the heat is left on indefinitely, what temperature will the inside of the house approach? Do not include units in your answer.

Question 64

The population of aphids on a rose plant increases at a rate proportional to the number present.In 3 days the population grew from 600 to 1400.How many days does it take for the population to get 10 times as large? Round to 2 decimal places.

Question 65

Is $y = \cos x + C _ { 1 } x + C _ { 2 }$ the general solution to the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \cos x$ ?

Question 66

Suppose there is a new kind of savings certificate that starts out paying 2% annual interest and increases the interest rate by 1% each additional year that the money is left on deposit.(Assume that interest is compounded continuously and that the interest rate increases continuously.)Write a differential equation that gives the rate of change in the balance B(t)at time t, and solve it assuming an initial deposit of $500.What is your equation?

A) $B=500 e^{0.02 t+0.005 t^{2}}$
B) $B=500 e^{0.02 t+0.01 t^{2}}$
C) $B=500 e^{0.02+0.01 t}$
D) $B=500 e^{0.03 t}$

Question 67

A bank account earns interest at a rate of 5% per year, compounded continuously.Money is deposited into the account in a continuous cash flow at a rate of $1000 per year.Use differential equations to find the amount of money in the bank account after 10 years, assuming an initial balance of $3000.Round to the nearest cent.

Question 68

Cesium 137 (Cs¹³⁷)is a short-lived radioactive isotope.It decays at a rate proportional to the amount of itself present and has a half-life of 30 years (i.e., the amount of Cs¹³⁷ remaining t years after A₀ mg of the radioactive isotope is released is given by $e^{-\left(\frac{\ln 2}{30}\right) t}$ ).As a result of its operations, a nuclear power plant releases Cs¹³⁷ at a rate of 0.12 mg per year.The plant began its operations in 1990, which we will designate as t = 0.Assume there is no other source of this particular isotope.Which of the following differential equations have a solution R(t), the amount (in mg)of Cs¹³⁷ in t years? (We are assuming R(0)= 0.)

A) $\frac{d R}{d t}=0.12+e^{-\left(\frac{\ln 2}{30}\right) R}$
B) $\frac{d R}{d t}=0.12-e^{-\left(\frac{\ln 2}{30}\right)}+R$
C) $\frac{d R}{d t}=0.12-\frac{\ln 2}{30} R$
D) $\frac{d R}{d t}=0.12 R-\frac{\ln 2}{30}$

Question 69

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding air.A detective discovers a corpse in an abandoned building, and finds its temperature to be 26°C.An hour later its temperature is 18°C.Assume that the air temperature is 6°C, that normal body temperature is 37°C, and that Newton's Law of Cooling applies to the corpse.How many hours has the corpse been dead at the moment it is discovered? Round to 2 decimal places.

Question 70

Cesium 137 (Cs¹³⁷)is a short-lived radioactive isotope.It decays at a rate proportional to the amount of itself present and has a half-life of 30 years (i.e., the amount of Cs¹³⁷ remaining t years after A₀ mg of the radioactive isotope is released is given by $e^{-\left(\frac{\ln 2}{30}\right) t}$ ).As a result of its operations, a nuclear power plant releases Cs¹³⁷ at a rate of 0.12 mg per year.The plant began its operations in 1990, which we will designate as t = 0.Assume there is no other source of this particular isotope.Which of the following integrals give the total amount of Cs¹³⁷ T years after the plant began operations?

A) $\int_{0}^{T} e^{\left(-\frac{0.12 \ln 2}{30}\right) t} d t$
B) $\int_{0}^{T} 0.12 e^{\left(-\frac{\ln 2}{30}\right) t} d t$
C) $\int_{0}^{0.12} \ t e^{\left(-\frac{\ln 2}{30}\right) t} d t$
D) $\left.\int_{0}^{T}\left(e^{\left(-\frac{\ln 2}{30}\right.}\right) t+0.12\right) d t$

Question 71

Mark all of the differential equations that are NOT separable.

A) $\frac { d y } { d t } = 3 t + 6 y$
B) $\frac { d y } { d x } = 6 x + 5 y ^ { 3 }$
C) $\frac { d y } { d x } = 5 e ^ { xy }$
D) $\frac { d y } { d x } = 6 e ^ { x + y }$

Question 72

The population of aphids on a rose plant increases at a rate proportional to the number present.In 5 days the population grew from 700 to 1100.Which of the following formulas give for the population of aphids at time t in days, where t = 0 is the day when there were 700 aphids?

A) $P=700 e^{5 t}$
B) $P=700 e^{\left(\frac{1}{5} \ln \frac{11}{7}\right) t}$
C) $P=700 e^{2000 t}$
D) $P=700 e^{\left(\frac{1}{5} \ln 400\right) t}$

Question 73

Suppose there is a new kind of savings certificate that starts out paying 3.5% annual interest and increases the interest rate by 1% each additional year that the money is left on deposit.(Assume that interest is compounded continuously and that the interest rate increases continuously.)Assuming an initial deposit of $1500, which of the following investment choices is better?

A)Investing the money at a fixed interest rate of 5% for 4 years.
B)Investing the money in the new kind of savings certificate for 4 years.

Question 74

Cesium 137 (Cs¹³⁷)is a short-lived radioactive isotope.It decays at a rate proportional to the amount of itself present and has a half-life of 30 years (i.e., the amount of Cs¹³⁷ remaining t years after A₀ mg of the radioactive isotope is released is given by ).As a result of its operations, a nuclear power plant releases Cs¹³⁷ at a rate of 0.14 mg per year.The plant began its operations in 1990, which we will designate as t = 0.Assume there is no other source of this particular isotope.Since Cs¹³⁷ poses a great health risk, the government says that the maximum amount of Cs¹³⁷ acceptable in the surrounding environment is 1 mg (spread over the surroundings).How many mg per year of the isotope can the station release and still be in compliance with the regulations? Round to 2 decimal places.

Question 75

Is $y = \frac { 1 } { 12 } x ^ { 4 } - \frac { 5 } { 2 } x ^ { 3 } + x ^ { 2 } + C _ { 1 } x + C _ { 2 }$ the general solution to the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } = x ^ { 2 } - 15 x + 2$ ?

Question 76

Consider the Hakosalo residence in Oulu, Finland.Assume that heat is lost from the house only through windows and the rate of change of temperature in °F/hr is proportional to the difference in temperature between the outside and the inside.The constant of proportionality is .Assume that it is 10°F outside constantly.On a Thursday at noon the temperature inside the house was 65°F and the heat was turned off until 5 pm.Use differential equations to find the temperature in the house at 5 pm.Round to one decimal place, and do not include units in your answer.

Question 77

Suppose there is a new kind of savings certificate that starts out paying 1.5% annual interest and increases the interest rate by 1% each additional year that the money is left on deposit.(Assume that interest is compounded continuously and that the interest rate increases continuously.)Which of the following differential equations gives the rate of change in the balance B(t)at time t?

A) $\frac{d B}{d t}=B(0.015+0.01 t)$
B) $\frac{d B}{d t}=B(0.015 t+0.01)$
C) $\frac{d B}{d t}=B(0.025) t$
D) $\frac{d B}{d t}=B(0.025+0.01 t)$

Question 78

Consider the Hakosalo residence in Oulu, Finland.Assume that heat is lost from the house only through windows and the rate of change of temperature in °F/hr is proportional to the difference in temperature between the outside and the inside.The constant of proportionality is $\frac { 1 } { 28 }$ .Assume that it is 10°F outside constantly.On a Thursday at noon the temperature inside the house was 65°F and the heat was turned off until 5 pm.Which of the following differential equations reflects the rate of change of the temperature in the house between noon and 5 pm?

A) $\frac { d y } { d t } = - \frac { 1 } { 28 } ( y - 10 )$ , $0 \leq t \leq 5$
B) $\frac { d y } { d t } = \frac { 1 } { 28 } ( y - 10 )$ , $0 \leq t \leq 5$
C) $\frac{d y}{d t}=\frac{1}{28} y-10$ , $0 \leq t \leq 5$
D) $\frac{d y}{d t}=-\frac{1}{28} y+10$ , $0 \leq t \leq 5$

Question 79

Solve the differential equation using separation of variables. $\frac { d y } { d x } = x y + 6 y + 2 x + 12$

A) $y = \frac { x ^ { 2 } y } { 2 } + \frac { 6 y ^ { 2 } } { 2 } + \frac { 2 x ^ { 2 } } { 2 } + 12 x + C$
B) $y = x y ^ { 2 } - 2 x ^ { 2 } + C$
C) $x = 5 , y = 0$
D) $y = C e ^ { \frac { x ^ { 2 } } { 2 } 6 x } - 2$

Question 80

Consider the Hakosalo residence in Oulu, Finland.Assume that heat is lost from the house only through windows and the rate of change of temperature in °F/hr is proportional to the difference in temperature between the outside and the inside.The constant of proportionality is $\frac{1}{31}$ .Assume that it is 10° F outside constantly.On a Thursday at noon the temperature inside the house was 65°F and the heat was turned off until 5 pm.At 5 pm the heat is turned on.The heater generates an amount of energy that would raise the inside temperature by 2°F per hour if there were no heat loss.Which of the following differential equations reflect what happens to the inside temperature after the heat is turned on?

A) $\frac{d y}{d t}=-\frac{1}{31} y+72$
B) $\frac{d y}{d t}=-\frac{1}{31} y+2$
C) $\frac{d y}{d t}=-\frac{1}{31} y+\frac{52}{31}$
D) $\frac{d y}{d t}=-\frac{1}{31} y+\frac{72}{31}$