Exam 9: Differential Equations

For what nonzero values of $k$ does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { tt } + 100 y = 0$ for all values of $A$ and $B$ ? a. $k = 10$ b. $k = - 100$ c. $k = - 10$ d. $k = 100$ e. $k = 1$

Which equation does the function $y = e ^ { - 6 t }$ satisfy?

Determine whether the differential equation is linear. $y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

An object with mass $m$ is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If $s ( t )$ is the distance dropped after $t$ seconds, then the speed is $v = s ^ { \prime } ( t )$ and the acceleration is $a = v ^ { \prime } ( t )$ . If $g$ is the acceleration due to gravity, then the downward force on the object is $m g - c v$ , where $c$ is a positive constant, and Newton's Second Law gives $m \frac { d v } { d t } = m g - c v$ . Find the limiting velocity.

Select a direction field for the differential equation $y ^ { \prime } = y ^ { 2 } - x ^ { 2 }$ from a set of direction fields labeled I-IV.

Suppose that a population grows according to a logistic model with carrying capacity 7,300 and $k = 0.04$ per year. Write the logistic differential equation for these data.

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.

Which of the following functions is a solution of the differential equation? $y ^ { tt } + 16 y ^ { t } + 64 y = 0$

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

Let $P ( t )$ be the performance level of someone learning a skill as a function of the training time $t$ . The graph of $P$ is called a learning curve. We propose the differential equation $\frac { d P } { d t } = r ( G - P ( t ) )$ as a reasonable model for learning, where $r$ is a positive constant. Solve it as a linear differential equation.

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 9 }$

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 6 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Choose the differential equation corresponding to this direction field.

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , r ( 0 ) = 3$

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

Solve the differential equation. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

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 $k y ^ { 1.02 }$ . If 7 such rabbits breed initially and the warren has 21 rabbits after 8 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).

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 $k y ^ { 103 }$ . If 3 such rabbits breed initially and the warren has 28 rabbits after 5 months, then when is doomsday?

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 2,000 inhabitants, 130 people have a disease at the beginning of the week and 1,100 have it at the end of the week. How long does it take for $80 \%$ of the population to be infected?