미분방정식 전체 문제

The population of Florida panthers in the southwestern tip of Florida has an estimated carrying capacity of $250$ and grows at a rate of $0.25\%$ per year and there must be $25$ for the population to survive. With an initial population of $30$ Florida panthers, how many years will it take to get the Florida panthers off the endangered species list (at least $100)?$

Assume that for a population, $K=1000$ and $\alpha =0.05.$ Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?

[T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999.$ Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached $7$ billion.

[T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999.$ Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached $7$ billion. Was logistic growth or Gompertz growth more accurate, considering world population reached $7$ billion on October $31,2011?$

When does population increase the fastest in the threshold logistic equation $P'(t)=rP\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)?$

When does population increase the fastest for the Gompertz equation $P(t)'=\alpha \ln \left(\frac{K}{P(t)}\right)P(t)?$

Using the three equations found in the previous problems, estimate the population in $2010$ (year $70$ after conservation). The real population measured at that time was $437.$ Which model is most accurate?

A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k.$ Set up the differential equation and solve for the velocity given an initial velocity of $0.$

Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior; does the velocity approach a value?)

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8$ m/s² and the proportionality constant is $4?$

A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant $k.$ Set up the differential equation and solve for the velocity.

Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8{\,\text{m/s}}^{2}$ and the proportionality constant is $4?$ Does it take more or less time than your initial estimate?

A car drives along a freeway, accelerating according to $a=5\sin (\pi t),$ where $t$ represents time in minutes. Find the velocity at any time $t,$ assuming the car starts with an initial speed of $60$ mph.

You throw a ball of mass $2$ kilograms into the air with an upward velocity of $8$ m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by $g=9.8{\,\text{m/s}}^{2}.$

You drop the same ball of mass $5$ kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of $3$ and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

A drug is administered to a patient every $24$ hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $0.2.$ If the patient needs a baseline level of $5$ mg to be in the bloodstream at all times, how large should the dose be?

A $1000$-liter tank contains pure water and a solution of $0.2$ kg salt/L is pumped into the tank at a rate of $1$ L/min and is drained at the same rate. Solve for total amount of salt in the tank at time $t.$

You boil water to make tea. When you pour the water into your teapot, the temperature is $100\text{°C.}$ After $5$ minutes in your $15\text{°C}$ room, the temperature of the tea is $85\text{°C.}$ Solve the equation to determine the temperatures of the tea at time $t.$ How long must you wait until the tea is at a drinkable temperature $(72\text{°C})?$

The human population (in thousands) of Nevada in $1950$ was roughly $160.$ If the carrying capacity is estimated at $10$ million individuals, and assuming a growth rate of $2\%$ per year, develop a logistic growth model and solve for the population in Nevada at any time (use $1950$ as time = 0). What population does your model predict for $2000?$ How close is your prediction to the true value of $1,998,257?$