Exam 9: Differential Equations

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 min?

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10$

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

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 $2000$ inhabitants, $110$ people have a disease at the beginning of the week and $1100$ have it at the end of the week. How long does it take for $70 \%$ of the population to be infected?

Suppose that a population grows according to a logistic model with carrying capacity $8,000$ and $k = 0.01$ per year. Choose the logistic differential equation for these data.

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

Find the solution of the differential equation $\frac { d y } { d x } = \frac { y ^ { 2 } } { x ^ { 4 } }$ that satisfies the initial condition $y ( 1 ) = 1$ .

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

Suppose that a population grows according to a logistic model with carrying capacity $6000$ and $k = 0.06$ per year. Write the logistic differential equation for these data.

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?

Use Euler's method with step size 0.25 to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = 6 x + y ^ { 2 } , \quad y ( 0 ) = 0$

Consider the differential equation $\frac { d P ( t ) } { d t } = 0.6 P \left( 1 - \frac { P } { 300 } \right) - c$ 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?

A tank contains $1050$ L of brine with $20$ kg of dissolved salt. Pure water enters the tank at a rate of $16$ L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after $20$ minutes?

Let $\frac { d P ( t ) } { d t } = 0.1 P \left( 1 - \frac { P } { 830 } \right) - \frac { 1810 } { 83 }$ . What are the equilibrium solutions?

Biologists stocked a lake with $700$ fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be $10800$ . 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.

Use Euler's method with step size 0.1 to estimate $y ( 0.5 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = y ( x + 2 ) , y ( 0 ) = 1$

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

The solution of the differential equation $y ^ { \prime } = y ^ { 3 } - 4 y$ satisfies the initial condition $y ( 0 ) = 0.2$ . Find the limit. $\lim _ { t \rightarrow \infty } y ( t )$

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