Exam 9: Differential Equations

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

Solve the initial-value problem. $r ^ { t } + 2 t r = r , r ( 0 ) = 5$

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?

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 $35 \mathrm {~min}$ ?

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 $k y ^ { 102 }$ . If 7 such rabbits breed initially and the warren has 21 rabbits after 8 months, then when is doomsday?

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

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }$

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?

$\text { Let } \frac { d P ( t ) } { d t } = 0.1 P \left( 1 - \frac { P } { 830 } \right) - \frac { 1,810 } { 83 } \text {. }$ $\text { What are the equilibrium solutions? }$

Solve the differential equation. $3 y y ^ {t } = 7 x$

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 7 x ^ { 6 } y = 6 x ^ { 6 }$

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

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

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

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.04 P - 0.0004 P ^ { 2 }$ where $t$ is measured in weeks. What is the carrying capacity? Select the correct answer.

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. $y ^ { t } = x ^ { 4 } e ^ { - \sin x } - y \cos x$

Consider a population $P = P ( t )$ with constant relative birth and death rates $a$ and $\beta$ , respectively, and a constant emigration rate $m$ , where $\alpha = 0.9 , \beta = 0.7$ and $m = 0.8$ . 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 $P ( 0 ) = 2200$ .

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 $35 \mathrm {~min}$ ? Select the correct answer.