Exam 9: Differential Equations

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

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

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, 140 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? Select the correct answer.

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

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.

Solve the differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

A curve passes through the point $( 4,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$ . What is the equation of the curve?

Solve the differential equation. $y ^ { \prime } = x ^ { 3 } e ^ { - \sin x } - y \cos x$

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.03 P - 0.0003 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity?

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 $\frac { d P } { d t } = r ( E - P ( t ) )$ as a reasonable model for learning, where $r$ is a positive constant. Solve it as a linear differential equation.

Consider a population $P = P ( t )$ with constant relative birth and death rates $a$ and $\beta$ , respectively, and a constant emigration rate $m$ , where $\alpha = 0.8 , \beta = 0.7$ and $m = 0.5$ . Then the rate of change of the population at time $t$ is modeled by the differential equation $\frac { d P } { d t } = k P - m$ where $k = \alpha - \beta$ Find the solution of this equation with the rate of change of the population at time $t = 6$ that satisfies the initial condition $P ( 0 ) = 2900$ .

Which equation does the function $y = e ^ { - 3 t }$ satisfy? Select the correct answer.

Which of the following functions are the constant solutions of the equation $\frac { d y } { d t } = y ^ { 4 } - y ^ { 3 } + 6 y ^ { 2 }$

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 the correct answer.

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

Solve the differential equation. $\frac { d u } { d t } = 15 + 3 u + 5 t + u t$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4480 } \right)$ For what values of $P$ is the population increasing?

Solve the differential equation. $y ^ { t } = x ^ { 5 } e ^ { - \sin x } - y \cos x$

Solve the initial-value problem. Select the correct answer. $r ^ { t } + 2 t r = r , \quad r ( 0 ) = 2$