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Chapter 1: Q15 E (page 5)

In Problems 13-16, write a differential equation that fits the physical description. The rate of change in the temperature T of coffee at time t is proportional to the difference between the temperature M of the air at time t and the temperature of the coffee at time t.

Short Answer

Expert verified

dTdt=kM-Twhere k is the constant of proportionality.

Step by step solution

01

Analyzing the given statement

By analyzing the given statement,

dTdt鈭�M-T, where T is the temperature of coffee at time t and M is the air temperature at time t.

02

Writing the differential equation

dTdt=kM-T, where k is the constant of proportionality.

Hence, dTdt=kM-T is the required differential equation.

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Most popular questions from this chapter

Spring Pendulum.Let a mass be attached to one end of a spring with spring constant kand the other end attached to the ceiling. Letlo be the natural length of the spring, and let l(t) be its length at time t. If胃(t) is the angle between the pendulum and the vertical, then the motion of the spring pendulum is governed by the system

l''(t)-l(t)胃'(t)-驳肠辞蝉胃(t)+km(l-lo)=0l2(t)胃''(t)+2l(t)l'(t)胃'(t)+gl(t)蝉颈苍胃(t)=0

Assume g = 1, k = m = 1, and lo= 4. When the system is at rest, l=lo+mgk=5.

a. Describe the motion of the pendulum when l(0)=5.5,l'(0)=0,胃(0)=0,胃'(0)=0.

b. When the pendulum is both stretched and given an angular displacement, the motion of the pendulum is more complicated. Using the Runge鈥揔utta algorithm for systems with h = 0.1 to approximate the solution, sketch the graphs of the length l and the angular displacement u on the interval [0,10] if l(0)=5.5,l'(0)=0,胃(0)=0.5,胃'(0)=0.

In Problems 15-24 , solve for Y(s), the Laplace transform of the solution ytto the given initial value problem.

y''+3y=t3;y0=0,y'0=0

The temperatureT(in units of 100 F) of a university classroom on a cold winter day varies with timet(in hours) asdTdt=1-T,ifheatingunitsisOn-T,ifheatingunitsisOFF.

T=0Suppose at 9:00 a.m., the heating unit is ON from 9-10 a.m., OFF from 10-11 a.m., ON again from 11 a.m.鈥搉oon, and so on for the rest of the day. How warm will the classroom be at noon? At 5:00 p.m.?

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

3xdxdt+4t=0,x(2)=-蟺

In Problems 13-16, write a differential equation that fits the physical description. The rate of change of the mass A of salt at time t is proportional to the square of the mass of salt present at time t.
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