Exam 6: Differential Equations

The half-life of the carbon isotope C-14 is approximately 5,715 years.If the amount left after 3,000 years is 1.8 g,what is the amount after 6,000 years? Round your answer to three decimal places. ​

Each of the following graphs is from a logistic function .Which one has the smallest value of b? ​

The logistic function models the growth of a population.Determine when the population reaches one-half of the maximum carrying capacity.Round your answer to three decimal places. ​

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

A container of hot liquid is placed in a freezer that is kept at a constant temperature of .The initial temperature of the liquid is .After 5 minutes,the liquid's temperature is .How much longer will it take for its temperature to decrease to ? Round your answer to two decimal places. ​

​Consider the differential equation ,for which the solution is g(x).​ ​Let y(x)be the particular solution that contains (0,1).Using Euler's Method with step size ,what is the estimate for ? ​

Solve the differential equation in to find f(x)in terms of x only.​

Find the orthogonal trajectories of the family ​ . ​

Find the orthogonal trajectories of the family . ​

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

Find the particular solution of the differential equation that satisfies the initial condition when ,where is the general solution. ​

Solve the differential equation . ​

A calf that weighs 60 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 650 pounds,find the time of sale using the model .Round your answer to two decimal places. ​

The logistic function models the growth of a population.Identify the value of k. ​

Sketch the slope field for the differential equation and use the slope field to sketch the solution that passes through the point . ​

Consider the differential equation ​,for which the solution is .Let . The particular solution is ​

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

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

​Consider the differential equation ​ ,for ​ only,with initial value . ​ Which of the following is the slope field for the general solution to the given differential equation?​ ​

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