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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

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Short Answer

Feedback System with Pooling Delay. Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure 5.3 on page 251. Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given by R1t=αV-V2t where α and V are positive constants and V2t is the volume of fluid in tank B at time t.

  1. If the outflow rate from tank B is constant and the flow rate from tank A into B is R2t=KV1t where K is a positive constant and V1t is the volume of fluid in tank A at time t, then show that this feedback system is governed by the system


b. Find a general solution for the system in part (a) when α=5min-1,V=20L,K=2min-1, and R3=10L/min.

c. Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as t+?

  1. The given system is proved as true. So, the systems are dV1dt=αV-V2t-KV1t and dV2dt=KV1t-R3.
  2. The general solutions of part (a) are V1t=Ae-tsin3t+Be-tcos3t+5 and V2=18-A-3Be-tsin3t+3A+Be-tcos3t5.
  3. The volume of tank A is 5 L/min and 18 L/min the volume of B is as t+.
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 when t+ we need for both roots of the

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.

Step 2: Evaluate the given equation

Given that, the volume of fluid in tank A is V1t and the volume of fluid in tank B is V2t:

R1 is the rate at which the fluid enters tank A.

R2 is the rate at which the fluid exits tank A and enters tank B.

R3 is the rate at which the fluid exits tank B.

And R1t=αV-V2t.

Given:R2t=KV1t .



The above equations can be rewritten as,

role="math" dV1dt=R1-R2=αV-V2t-KV1tdV2dt=R2-R3=KV1t-R3

Given, α=5min-1,V=20L,K=2min-1, and R3=10L/min.

Referring to part (a):



Rewrite the system in operator form:


Substitute the values in equations (1) and (2).



Step 3: Solve the equations

Multiply D on equation (3) and multiply 5 on equation (4). Then, subtract them together.


Since the auxiliary equation to the corresponding homogeneous equation is:

r2+2r+10=0 .



So, the roots are r = - 1 + 3i and r = - 1 - 3i.

Then, the general solution of y is V1ht=Ae-tsin3t+Be-tcos3t......(6)

Let us assume that, V1pt=C......(7)

Substitute equation (7) in equation (5).


Substitute the value of C in equation (7).


So, V1t=Ae-tsin3t+Be-tcos3t+5

Step 4: Substitution method

Now substitute equation (8) in equation (3).



Hence, V2=18-A-3Be-tsin3t+3A+Be-tcos3t5

Step 5: limit method

To find: limtV1 and limtV2.

Referring to part (b):


Implement the limits on equations (8) and (9).

role="math" localid="1664084322908" limtV1=limtAe-tsin3t+Be-tcos3t+5=5

role="math" localid="1664084386450" limtV2=limt18-A-3Be-tsin3t+3A+Be-tcos3t5=18

Therefore, the solution is found.

Most popular questions for Math Textbooks

Sticky Friction. An alternative for the damping friction model F = -by′ discussed in Section 4.1 is the “sticky friction” model. For a mass sliding on a surface as depicted in Figure 5.18, the contact friction is more complicated than simply -by′. We observe, for example, that even if the mass is displaced slightly off the equilibrium location y = 0, it may nonetheless remain stationary due to the fact that the spring force -ky is insufficient to break the static friction’s grip. If the maximum force that the friction can exert is denoted by m, then a feasible model is given by

\({{\bf{F}}_{{\bf{friction}}}}{\bf{ = }}\left\{ \begin{array}{l}{\bf{ky,if}}\left| {{\bf{ky}}} \right|{\bf{ < }}\mu {\bf{andy' = 0}}\\\mu {\bf{sign(y),if}}\left| {{\bf{ky}}} \right| \ge {\bf{0andy' = 0}}\\ - \mu {\bf{sign(y'),ify'}} \ne 0.\end{array} \right.\)

(The function sign (s) is +1 when s 7 0, -1 when s 6 0, and 0 when s = 0.) The motion is governed by the equation (16) \({\bf{m}}\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - ky + }}{{\bf{F}}_{{\bf{friction}}}}\)Thus, if the mass is at rest, friction balances the spring force if \(\left| {\bf{y}} \right|{\bf{ < }}\frac{\mu }{{\bf{k}}}\)but simply opposes it with intensity\(\mu \)if\(\left| {\bf{y}} \right| \ge \frac{\mu }{{\bf{k}}}\). If the mass is moving, friction opposes the velocity with the same intensity\(\mu \).

  1. Taking m =\(\mu \) = k = 1, convert (16) into the firstorder system y′ = v (17)\({\bf{v' = }}\left\{ \begin{array}{l}{\bf{0,if}}\left| {\bf{y}} \right|{\bf{ < 1andv = 0}}{\bf{.}}\\{\bf{ - y + sign(y),if}}\left| {\bf{y}} \right| \ge {\bf{1andv = 0}}\\{\bf{ - y - sign(v),ifv}} \ne 0\end{array} \right.\) ,
  2. Form the phase plane equation for (17) when v ≠ 0 and solve it to derive the solutions\({{\bf{v}}^{\bf{2}}}{\bf{ + (y \pm 1}}{{\bf{)}}^{\bf{2}}}{\bf{ = c}}\).where the plus sign prevails for v>0 and the minus sign for v<0.
  3. Identify the trajectories in the phase plane as two families of concentric semicircles. What is the centre of the semicircles in the upper half-plane? The lower half-plane?
  4. What are the critical points for (17)?
  5. Sketch the trajectory in the phase plane of the mass released from rest at y = 7.5. At what value for y does the mass come to rest?

Rigid Body Nutation. Euler’s equations describe the motion of the principal-axis components of the angular velocity of a freely rotating rigid body (such as a space station), as seen by an observer rotating with the body (the astronauts, for example). This motion is called nutation. If the angular velocity components are denoted by x, y, and z, then an example of Euler’s equations is the three-dimensional autonomous system

\(\begin{array}{l}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = yz}}\\\frac{{{\bf{dy}}}}{{{\bf{dt}}}}{\bf{ = - 2xz}}\\\frac{{{\bf{dz}}}}{{{\bf{dt}}}}{\bf{ = xy}}\end{array}\)

The trajectory of a solution x(t),y(t), z(t) to these equations is the curve generated by the points (x(t), y(t), z(t) ) in xyz-phase space as t varies over an interval I.

(a) Show that each trajectory of this system lies on the surface of a (possibly degenerate) sphere centered at the origin (0, 0, 0).[Hint: Compute\(\frac{{\bf{d}}}{{{\bf{dt}}}}{\bf{(}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}{\bf{ + }}{{\bf{z}}^{\bf{2}}}{\bf{)}}\)What does this say about the magnitude of the angular velocity vector?

(b) Find all the critical points of the system, i.e., all points\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{y}}_{\bf{o}}}{\bf{,}}{{\bf{z}}_{\bf{o}}}{\bf{)}}\) such that \({\bf{x(t) = }}{{\bf{x}}_{\bf{o}}}{\bf{,y(t) = }}{{\bf{y}}_{\bf{o}}}{\bf{,z(t) = }}{{\bf{z}}_{\bf{o}}}\) is a solution. For such solutions, the angular velocity vector remains constant in the body system.

(c) Show that the trajectories of the system lie along the intersection of a sphere and an elliptic cylinder of the form\({{\bf{y}}^{\bf{2}}}{\bf{ + 2}}{{\bf{x}}^{\bf{2}}}{\bf{ = C}}\) for some constant C. [Hint: Consider the expression for dy/dx implied by Euler’s equations.]

(d) Using the results of parts (b) and (c), argue that the trajectories of this system are closed curves. What does this say about the corresponding solutions?

(e) Figure 5.19 displays some typical trajectories for this system. Discuss the stability of the three critical points indicated on the positive axes.


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