Exam 9: Differential Equations

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?

A curve passes through the point $( 6,2 )$ and has the property that the slope of the curve at every point $P$ is 4 times the $\mathrm { y }$ -coordinate $P$ . What is the equation of the curve?

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 11 }$

Solve the differential equation. $4 \frac { d w } { d t } + 5 e ^ { t + w } = 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4360 } \right)$ For what values of $P$ is the population increasing? Select the correct answer

Solve the differential equation. $3 y y ^ { t } = 5 x$

Suppose that a population grows according to a logistic model with carrying capacity 7,000 and $k = 0.05$ per year. Choose the logistic differential equation for these data.

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \cos x , y ( \pi ) = 0$

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.

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 6,000 inhabitants, 160 people have a disease at the beginning of the week and 1,500 have it at the end of the week. How long does it take for $60 \%$ of the population to be infected?

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

A population is modeled by the differential equation $\frac { d p } { d t } = 1.8 P \left( 1 - \frac { P } { 5120 } \right)$ For what values of $P$ is the population decreasing?

The population of the world was about $5.3$ billion in 1990 . Birth rates in the 1990 s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take $k$ to be an estimate of the initial relative growth rate.)

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

A function $y ( t )$ satisfies the differential equation $\frac { d y } { d t } = y ^ { 4 } - 9 y ^ { 3 } + 20 y ^ { 2 }$ . What are the constant solutions of the equation?

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.05 P - 0.0005 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity? Select the correct answer.

Consider the differential equation $\frac { d P ( t ) } { d t } = 0.4 P \left( 1 - \frac { P } { 900 } \right) - c$ as a model for a fish population, where $t$ is measured in weeks and $c$ is a constant. For what values of $c$ does the fish population always die out?

Solve the initial-value problem. $x y ^ { \prime } - y = x \ln x , y ( 1 ) = 6$

Choose the differential equation corresponding to this direction field.t

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $\frac { d A } { d t } = 5 A ( 1 - 0.0003 A ) - 0.03 A L$ $\frac { d L } { d t } = - 0.6 L + 0.0003 A L$ Find the equilibrium solution.