Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 251
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.


Short Answer

In Problem 36, if a small furnace that generates 1000 Btu/hr is placed in zone B, determine the coldest it would eventually get in zone B has a heat capacity of 2°F per thousand Btu.

Therefore, the cold of zone B is eventually got 4601141.8°F.

See the step by step solution

Step by Step Solution

Step 1: General form

Elimination Procedure for 2 x 2 Systems:

To find a general solution for the system


Where L1,L2,L3, and L4 are polynomials in D=ddt:

  1. Make sure that the system is written in operator form.
  2. Eliminate one of the variables, say, y, and solve the resulting equation for x(t). If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.
  3. (Shortcut) If possible, use the system to derive an equation that involves y(t) but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for x(t) into this equation to get a formula for y(t). The expressions for x(t), and y(t) give the desired general solution.
  4. Eliminate x from the system and solve for y(t). [Solving for y(t) gives more constants- twice as many as needed.]
  5. Remove the extra constants by substituting the expressions for x(t) and y(t) into one or both of the equations in the system. Write the expressions for x(t) and y(t) in terms of the remaining constants.

Vieta’s formulas for finding roots:

For function y(t) to be bounded, we need both roots of the auxiliary equation to be non-positive if they are reals and, if they are complex, then the real part has to be non-positive. In other words,

  1. If r1,r2R, then r1·r20,r1+r20,
  2. If r1,r2=α±βi,β0, then α=r1+r220.

Section 3.3: Heat transfer, it is modelled by the following equation dTdt=KMt-Tt+Ht+Ut where 1K is the time constant for the building given in hours, Mt is the outside temperature, Tt is the inside temperature, Ht is the heating in the building, Ut is the cooling.

Step 2: Evaluate the given equation

Given that, 1000 Btu/hr is placed in zone B.

Referring to problem 36:

Only zone A is heated by a furnace, which generates 80,000 Btu/hr.

The heat capacity of zone A is 14°F per thousand Btu.

The time constant for heat transfer between zone A and the outside is 4 hours, between the unheated zone B and the outside is 5 hours, and between the two zones is 2 hours.

Let x(t) be denoted as the temperature in A at time t and y(t) be denoted as the temperature in B at time t.

Using the given information create the system of equation.

Then, dxdt=12yt-xt-xt4+20and dydt=12xt-yt-yt5+2

The above equations can be rewritten as,

role="math" localid="1664042606654" dxdt=12yt-xt-xt4+204dxdt=2yt-xt-xt+80=2yt-2xt+80-xt=2yt-3xt+80



Rewrite the system in operator form:


Step 3: Solve the equations

Multiply 5 on equation (3) and multiply 4D + 3 on equation (4). Then, add them together to get.



Since the auxiliary equation to the corresponding homogeneous equation is:




So, the roots are r=-29+40140 and r=-29-40140.

Then, the general solution of y is yht=Ae-29+40140t+Be-29-40140t......(6)

Let us assume that, ypt=C......(7)

Substitute equation (7) in equation (5).


Substitute the value of C in equation (7).


So, the general solution is yt=Ae-29+40140t+Be-29-40140t+46011......(8)

Step 4: limit method

To find: limtx.

Implement the limits on equation (8).

role="math" localid="1664043341919" limtyt=limtAe-29+40140t+Be-29-40140t+46011=46011

So, the solution is founded

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.