1. Assume that the rate at which a population of a country grows is proportional to the total population at that timewith a constant immigration rate of r > 0. Write a differential equation used to find the population P(t). (You do notneed to solve this differential equation.)2. A can of soda at 40°F is placed into a room where the temperature is 70°F. Assume that Newton's Law ofCooling/Warming applies: The rate of cooling/warming is proportional to the difference between the currenttemperature and the ambient temperature. Let T represent the current temperature of the soda at time, t, inminutes. Write an initial-value problem used to find the temperature of the soda. (You do not need to solve thisinitial-value problem.)3. A 10 gallon tank is filled with 10 gallons of water in which 3 pounds of salt is dissolved. A mixture containing asolution with 1 pound per gallon begins flowing into the tank at a rate of 2 gal/min. Simultaneously, a drain isopened at the bottom of the tank allowing the solution to leave the tank at a rate of 2 gal/min. Write an initial-valueproblem used to find the amount of salt in the tank. (You do not need to solve this initial-value problem.)

Note: According to bartleby we have to answer only first question please upload the question…

Assume that the rate at which a population of a country grows is proportional to the total population at that time with a constant immigration rate of r > 0. Write a differential equation used to find the population P(t). (You do not need to solve this differential equation.)

Assume that the rate at which a population of a country grows is proportional to the total population at that time with a constant immigration rate of r > 0. Write a differential equation used to find the population P(t). (You do not need to solve this differential equation.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

1. Assume that the rate at which a population of a country grows is proportional to the total population at that time
with a constant immigration rate of r > 0. Write a differential equation used to find the population P(t). (You do not
need to solve this differential equation.)
2. A can of soda at 40°F is placed into a room where the temperature is 70°F. Assume that Newton's Law of
Cooling/Warming applies: The rate of cooling/warming is proportional to the difference between the current
temperature and the ambient temperature. Let T represent the current temperature of the soda at time, t, in
minutes. Write an initial-value problem used to find the temperature of the soda. (You do not need to solve this
initial-value problem.)
3. A 10 gallon tank is filled with 10 gallons of water in which 3 pounds of salt is dissolved. A mixture containing a
solution with 1 pound per gallon begins flowing into the tank at a rate of 2 gal/min. Simultaneously, a drain is
opened at the bottom of the tank allowing the solution to leave the tank at a rate of 2 gal/min. Write an initial-value
problem used to find the amount of salt in the tank. (You do not need to solve this initial-value problem.)

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According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T - Tm) where k is a positive constant. In this exercise, you will verify that T = Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Im T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T – Tm). - -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes