Exam 9: Differential Equations

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 3,000 inhabitants, 120 people have a disease at the beginning of the week and 1,600 have it at the end of the week. How long does it take for $40 \%$ of the population to be infected? Select the correct answer.

Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose that a roast turkey is taken from an oven when its temperature has reached $160 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $60 ^ { \circ } \mathrm { F }$ . If $u ( t )$ is the temperature of the turkey after $t$ minutes, then Newton's Law of Cooling implies that $\frac { d u } { d t } = k ( u - 60 )$ This could be solved as a separable differential equation. Another method is to make the change of variable $y = u - 60$ . If the temperature of the turkey is $150 ^ { \circ } \mathrm { F }$ after half an hour, what is the temperature after $35 \mathrm {~min}$ ?

A curve passes through the point $( 10,8 )$ and has the property that the slope of the curve at every point $P$ is 2 times the $\mathrm { y }$ -coordinate $P$ . What is the equation of the curve?

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct statement.

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the differential equation. $y ^ { t } = \frac { 7 x ^ { 6 } y } { \ln y }$

A sum of $\$ 2170$ is invested at $11 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,700 . The number of fish tripled in the first year. Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after $t$ years.

Let $c$ be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 1.03 }$ . If 3 such rabbits breed initially and the warren has 28 rabbits after 5 months, then when is doomsday?

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,700 . The number of fish tripled in the first year. Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after $t$ years.

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 75 cells. Find the number of cells after 2 hours.

$\text { Choose the differential equation corresponding to this direction field. }$

Solve the differential equation. Select the correct answer. $x ^ { 2 } \frac { d y } { d x } - y = 10 x ^ { 3 } e ^ { - 1 / x }$

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

Find the solution of the initial-value problem and use it to find the population when $t = 10$ . $\frac { d P } { d t } = 0.4 P \left( 1 - \frac { P } { 2000 } \right) , P ( 0 ) = 100$

Let $c$ be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 1.05 }$ . If 4 such rabbits breed initially and the warren has 23 rabbits after 6 months, then when is doomsday?

A certain small country has $\$ 20$ billion in paper currency in circulation, and each day $\$ 70$ million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let $x = x ( t )$ denote the amount of new currency in circulation at time $t$ with $x ( 0 ) = 0$ . Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).

Suppose that a population grows according to a logistic model with carrying capacity 2,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

In the circuit shown in Figure, a generator supplies a voltage of $E ( t ) = 20 \sin 60 t$ volts, the inductance is $2 H$ , the resistance is $40 \Omega$ , and $I ( 0 ) = 2 A$ . Find the current $0.2 \mathrm {~s}$ after the switch is closed. Round your answer to two decimal places.