Question 1

Solve the Bernoulli differential equation  .

Accepted Answer

The answer of Solve the Bernoulli differential equation  .
...

Question 2

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​  ​  ​

Accepted Answer

The answer of Sketch a few solutions of the differential...

Question 3

Suppose that the population (in millions) of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003. Find the exponential growth model  for the population by letting  correspond to 2000. Round your answer to four decimal places. ​

Accepted Answer

The answer of Suppose that the population (in millions) of...

Question 4

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places. ​

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years. To double the money, $A = 2P$, so $2P = Pe^{rt}$. Simplifying, we get $2 = e^{rt}$. Substituting $r = 0.06$ (since 6% as a decimal is 0.06), we solve for $t$: $2 = e^{0.06t}$. Taking the natural logarithm of both sides gives $ln(2) = 0.06t$, and solving for $t$ gives $t = \frac{ln(2)}{0.06}$, which approximately equals 11.55 years.

Question 5

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is  years, what is the annual rate? Round your answer to two decimal places. ​

Accepted Answer

The answer of The initial investment in a savings account...

Question 6

Use integration to find a general solution of the differential equation. ​  ​

Accepted Answer

The answer of Use integration to find a general solution...

Question 7

Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where the air resistance is proportional to the velocity. Round numerical answers in your answer to two places. ​
(i) Write the velocity as a function of time if the object's velocity after 4 seconds is 2.33 feet per second.
​
(ii) What is the limiting value of the velocity function?
​

Accepted Answer

The answer of Assume an object weighing 7 pounds is...

Question 8

Find an equation of the graph that passes through the point  and has the slope  . ​

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The answer of Find an equation of the graph that...

Question 9

Find the function  passing through the point  with the first derivative  . ​

Accepted Answer

The answer of Find the function  passing through the...

Question 10

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ​  ,  ​

Accepted Answer

The answer of Use Euler's Method to make a table...

Question 11

Find the logistic equation that satisfies the following differential equation and initial condition. ​  ,  ​

Accepted Answer

The answer of Find the logistic equation that satisfies the...

Question 12

A calf that weighs 75 pounds at birth gains weight at the rate  where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model  . Round your answer to two decimal places. ​

Accepted Answer

The answer of A calf that weighs 75 pounds at...

Question 13

Solve the differential equation. ​  ​

Accepted Answer

The answer of Solve the differential equation. ​  ​
...

Question 14

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Accepted Answer

The growth of the wolf population can be modeled using the logistic growth equation, which is given by $ P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}} $, where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $P_0$ is the initial population, and $r$ is the growth rate. Given $K = 125$, $P_0 = 20$, and $P(2) = 35$, we first need to solve for $r$. Using $P(2) = 35$, we get $35 = \frac{125}{1 + \left(\frac{125 - 20}{20}\right)e^{-2r}}$. Solving this equation for $r$ gives us the growth rate. Once $r$ is known, we use the logistic growth equation to solve for $t$ when $P(t) = 80$, which gives us $t \approx 6.259$ years.

Question 15

Use the differential equation  and its slope field to find the slope at the point  .  ​

Accepted Answer

The answer of Use the differential equation  and its...

Question 16

Select from the choices below the slope field for the differential equation. ​  ​

Accepted Answer

The answer of Select from the choices below the slope...

Question 17

Find the particular solution of the differential equation  that satisfies the boundary condition  . ​

Accepted Answer

The answer of Find the particular solution of the differential...

Question 18

Match the logistic differential equation and initial condition with the graph of its solution shown below. ​  ​

Accepted Answer

The answer of Match the logistic differential equation and initial...

Question 19

A 300-gallon tank is half full of distilled water. At time  , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.

Accepted Answer

The answer of A 300-gallon tank is half full of...

Question 20

Solve the first order linear differential equation. ​  ​

Accepted Answer

The answer of Solve the first order linear differential equation....

Quiz 6: Differential Equations

Solve the Bernoulli differential equation .

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Suppose that the population (in millions) of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003. Find the exponential growth model for the population by letting correspond to 2000. Round your answer to four decimal places.

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.

Use integration to find a general solution of the differential equation.

Find an equation of the graph that passes through the point and has the slope .

Find the function passing through the point with the first derivative .

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ,

Find the logistic equation that satisfies the following differential equation and initial condition. ,

A calf that weighs 75 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places.

Solve the differential equation.

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Use the differential equation and its slope field to find the slope at the point .

Select from the choices below the slope field for the differential equation.

Find the particular solution of the differential equation that satisfies the boundary condition .

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Solve the first order linear differential equation.

Quiz 6: Differential Equations

Solve the Bernoulli differential equation .

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​ ​ ​

Suppose that the population (in millions) of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003. Find the exponential growth model for the population by letting correspond to 2000. Round your answer to four decimal places. ​

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places. ​

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places. ​

Use integration to find a general solution of the differential equation. ​ ​

Find an equation of the graph that passes through the point and has the slope . ​

Find the function passing through the point with the first derivative . ​

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ​ , ​

Find the logistic equation that satisfies the following differential equation and initial condition. ​ , ​

A calf that weighs 75 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places. ​

Solve the differential equation. ​ ​

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Use the differential equation and its slope field to find the slope at the point . ​

Select from the choices below the slope field for the differential equation. ​ ​

Find the particular solution of the differential equation that satisfies the boundary condition . ​

Match the logistic differential equation and initial condition with the graph of its solution shown below. ​ ​

Solve the first order linear differential equation. ​ ​

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Suppose that the population (in millions) of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003. Find the exponential growth model for the population by letting correspond to 2000. Round your answer to four decimal places.

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.

Use integration to find a general solution of the differential equation.

Find an equation of the graph that passes through the point and has the slope .

Find the function passing through the point with the first derivative .

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ,

Find the logistic equation that satisfies the following differential equation and initial condition. ,

A calf that weighs 75 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places.

Solve the differential equation.

Use the differential equation and its slope field to find the slope at the point .

Select from the choices below the slope field for the differential equation.

Find the particular solution of the differential equation that satisfies the boundary condition .

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Solve the first order linear differential equation.