Differential Equations: An Introduction to Modern Methods and Applications
Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Chapter 3.3, Problem 31P

Obtaining exact, or approximate, expressions for eigenvalues and eigenvectors in terms of the model parameters is often useful for understanding the qualitative behavior of solutions to a dynamical system. We illustrate using Example 1 in Section 3.2 .

(a) Show that the general solution of Eqs. ( 5 ) and ( 6 ) in section 3.2 can be represented as

u = c 1 x 1 ( t ) + c 2 x 2 ( t ) + u ^ (i)

where u ^ is the equilibrium solution (see Problem 13 , Section 3.2 ) to the system and { x 1 ( t ) , x 2 ( t ) } is a fundamental set of solutions to the homogeneous equation

x ' = ( ( k 1 + k 2 ) k 2 ε k 2 ε k 2 ) x = K x

(b) Assuming that 0 < ε 1 (i.e., ε is positive and small relative to unity), show that approximations to the eigenvalues of K are

λ 1 ( ε ) ( k 1 + k 2 ) ε k 2 2 k 1 + k 2 and λ 2 ( ε ) ε k 1 k 2 k 1 + k 2 .

(c) Show that approximations to the corresponding eigenvectors are

v 1 ( ε ) = ( 1 , ε k 2 / ( k 1 + k 2 ) ) T and v 2 ( ε ) = ( k 2 / ( k 1 + k 2 ) , 1 ) T .

(d) Use the approximations obtained in parts (b) and (c) and the equilibrium solution u ^ to write down an approximation to the general solution (i). Assuming that nominal values for k 1 and k 2 are near unity, and that 0 < ε 1 , sketch a qualitatively accurate phase portrait for Eqs. ( 5 ) and ( 6 ) in Section 3.2 . Compare your sketch with the phase portrait in Figure 3.2.4 , Section 3.2 .

(e) Show that when t is large and the system is in the quasi-steady state,

u 1 k 1 k 1 + k 2 T a + k 2 k 1 + k 2 u 2 ,

u 2 ( u 20 T a ) exp ( ε k 1 k 2 k 1 + k 2 t ) + T a ,

where u 2 ( 0 ) = u 20 . Now let ε 0 . Interpret the results. Compare with the results of Problem 14 in Section 3.2 .

(f) Give a physical explanation of the significance of the eigenvalues on the dynamical behavior of the solution (i) . In particular, relate the eigenvalues to fast and slow temporal changes in the state of the system. What implications does the value of ε have with respect to the design of the green-house/rockbed system? Explain, giving due consideration to local climatic conditions and construction costs.

Example 1 in Section 3.2

Consider the schematic diagram of the greenhouse/rockbed system in Figure 3.2.1 . The rockbed, consisting of rocks ranging in size from 2 to, is loosely packed so that air can easily pass through the void space between the rocks. The rockbed, and the underground portion of the air ducts used to circulate air through the system, are thermally insulated from the surrounding soil.

Rocks are a good material for storing heat since they have a high energy-storage capacity, are inexpensive, and have a long life.

d u 1 d t = ( k 1 + k 2 ) u 1 + k 2 u 2 + k 1 T a , ( 5 )

d u 2 d t = ε k 2 u 1 ε k 2 u 2 , ( 6 )

u = ( u 1 u 2 ) , b = ( 14 0 ) , K = ( 13 8 3 4 1 4 1 4 ) ( 13 )

d u d t = Ku + b ( 14 )

Using vector notation, the initial conditions are expressed as

u ( 0 ) = u 0 = ( u 10 u 20 ) ( 15 )

Chapter 3.3, Problem 31P, Obtaining exact, or approximate, expressions for eigenvalues and eigenvectors in terms of the model

Section 3.2

Find the equilibrium solution, or critical point, of Eqs. ( 5 ) and ( 6 ) of Example 1 .

d u 1 d t = ( k 1 + k 2 ) u 1 + k 2 u 2 + k 1 T a ( 5 )

d u 2 d t = ε k 2 u 1 ε k 2 u 2 ( 6 )

In the limiting case, ε 0 , Eqs. ( 5 ) and ( 6 ) of Example 1 reduce to the partially coupled system

d u 1 d t = ( k 1 + k 2 ) u 1 + k 2 u 2 + k 1 T a ( 5 )

d u 2 d t = ε k 2 u 1 ε k 2 u 2 ( 6 )

d u 1 d t = ( k 1 + k 2 ) u 1 + k 2 u 2 + k 1 T a , ( i )

d u 2 d t = 0 . ( i i )

Thus, if initial conditions u 1 ( 0 ) = u 10 and u 2 ( 0 ) = u 20 are prescribed, Eq. ( i i ) implies that u 2 ( t ) = u 20 for all t 0 . Therefore Eq. ( i ) reduces to a first order equation with one dependent variable,

d u 1 d t = ( k 1 + k 2 ) u 1 + k 2 u 20 + k 1 T a . ( i i i )

( a ) Find the critical point (equilibrium solution) of Eq. ( i i i ) and classify it as asymptotically stable or unstable. Then draw the phase line, and sketch several graphs of solutions in the t u 1 p l a n e .

( b ) Find the solution of Eq. ( i i i ) subject to the initial condition u 1 ( 0 ) = u 10 and use it to verify the qualitative results of part ( a ) .

( c ) What is the physical interpretation of setting ε = 0 ? Give a physical interpretation of the equilibrium solution found in part ( a ) .

( d ) What do these qualitative results imply about the sizing of the rock storage pile in combination with temperatures that can be achieved in the rock storage pile during the daytime?

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Chapter 3 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

Ch. 3.1 - Solving Linear Systems. In each of Problems 1...Ch. 3.1 - Solving Linear Systems. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - Eigenvalues and Eigenvectors. In each of Problems ...Ch. 3.1 - In each of Problems through : Find the...Ch. 3.1 - In each of Problems through : Find the...Ch. 3.1 - In each of Problems 33 through 36: Find the...Ch. 3.1 - In each of Problems through : Find the...Ch. 3.1 - If , derive the result in Eq. for . …...Ch. 3.1 - Show that =0 is an eigenvalue of the matrix A if...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Writing Systems in Matrix Form. In each of...Ch. 3.2 - Show that the functions and are solutions of...Ch. 3.2 - (a) Show that the functions x(t)=et(2cos2tsin2t)...Ch. 3.2 - Show that is solution of the...Ch. 3.2 - (a) Show that x=et(2t1t1)+(6t+22t1) issolution of...Ch. 3.2 - Find the equilibrium solution, or critical point,...Ch. 3.2 - Prob. 14PCh. 3.2 - In each of Problems through : Find the...Ch. 3.2 - In each of Problems through : Find the...Ch. 3.2 - In each of Problems 15 through 20: (a) Find the...Ch. 3.2 - In each of Problems 15 through 20: (a) Find the...Ch. 3.2 - In each of Problems 15 through 20: (a) Find the...Ch. 3.2 - In each of Problems through : Find the...Ch. 3.2 - Second Order Differential Equations. In Problems...Ch. 3.2 - Second Order Differential Equations. In Problems...Ch. 3.2 - Second Order Differential Equations. In Problems...Ch. 3.2 - Second Order Differential Equations. In Problems...Ch. 3.2 - In each of Problems 25 and 26, transform the given...Ch. 3.2 - In each of Problems 25 and 26, transform the given...Ch. 3.2 - Applications. Electric Circuits. The theory of...Ch. 3.2 - Applications. Electric Circuits. The theory of...Ch. 3.2 - Applications. Electric Circuits. The theory of...Ch. 3.2 - Mixing Problems. Each of the tank shown in...Ch. 3.2 - Consider two interconnected tanks similar to those...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. In each of problems...Ch. 3.3 - General Solutions of Systems. 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In each of...Ch. 3.3 - Second order Equations. For Problems through...Ch. 3.3 - Second order Equations. For Problems through...Ch. 3.3 - Second order Equations. For Problems through...Ch. 3.3 - Second order Equations. For Problems through...Ch. 3.3 - Second order Equations. For Problems through...Ch. 3.3 - Second order Equations. For Problems 25 through...Ch. 3.3 - Obtaining exact, or approximate, expressions for...Ch. 3.3 - Electric Circuits. Problem 32 and 33 are concerned...Ch. 3.3 - Electric Circuits. Problem and are concerned...Ch. 3.3 - Dependence on a Parameter. Consider the system...Ch. 3.4 - General Solutions of Systems. In each of Problems...Ch. 3.4 - General Solutions of Systems. In each of Problems ...Ch. 3.4 - General Solutions of Systems. In each of Problems...Ch. 3.4 - General Solutions of Systems. In each of Problems ...Ch. 3.4 - General Solutions of Systems. In each of Problems...Ch. 3.4 - General Solutions of Systems. In each of Problems ...Ch. 3.4 - In each of Problems through, find the solution of...Ch. 3.4 - In each of Problems through, find the solution of...Ch. 3.4 - In each of Problems 7 through 10, find the...Ch. 3.4 - In each of Problems through, find the solution of...Ch. 3.4 - Phase Portraits and component Plots. In each of...Ch. 3.4 - Phase Portraits and component Plots. In each of...Ch. 3.4 - Dependence on a Parameter. In each of Problems ...Ch. 3.4 - Dependence on a Parameter. In each of Problems ...Ch. 3.4 - Dependence on a Parameter. In each of Problems ...Ch. 3.4 - Dependence on a Parameter. In each of Problems ...Ch. 3.4 - Dependence on a Parameter. In each of Problems ...Ch. 3.4 - Dependence on a Parameter. In each of Problems 13...Ch. 3.4 - Dependence on a Parameter. In each of Problems 13...Ch. 3.4 - Dependence on a Parameter. In each of Problems 13...Ch. 3.4 - Applications. Consider the electric circuit shown...Ch. 3.4 - Applications. The electric circuit shown in...Ch. 3.4 - Applications. In this problem, we indicate how to...Ch. 3.5 - General Solution and Phase Portraits. In each of...Ch. 3.5 - General Solutions and Phase Portraits. In each of...Ch. 3.5 - General Solutions and Phase Portraits. In each of...Ch. 3.5 - General Solutions and Phase Portraits. In each of...Ch. 3.5 - General Solutions and Phase Portraits. In each of...Ch. 3.5 - General Solutions and Phase Portraits. In each of...Ch. 3.5 - In each of Problems 7 through , find the solution...Ch. 3.5 - In each of Problems 7through 12, find the solution...Ch. 3.5 - In each of Problems 7 through , find the solution...Ch. 3.5 - In each of Problems 7 through , find the solution...Ch. 3.5 - In each of Problems 7 through , find the solution...Ch. 3.5 - In each of Problems 7 through , find the solution...Ch. 3.5 - Consider again the electric circuit in Problem 22...Ch. 3.5 - Trace Determinant Plane. Show that the solution of...Ch. 3.5 - Consider the linear system , where and are real...Ch. 3.5 - Continuing Problem 15, Show that the critical...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem 1 through 6: a)...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem 1 through 6: a)...Ch. 3.6 - For each of the systems in Problem 7 through 12:...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem 13 through 20:...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem 13 through 20:...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem through : Find...Ch. 3.6 - For each of the systems in Problem 13 through 20:...Ch. 3.6 - Consider the system in Example . Draw a component...Ch. 3.6 - In this problem we indicate how to find the...Ch. 3.6 - Prob. 23PCh. 3.6 - An asymptotically stable limit cycle is a closed...Ch. 3.6 - A model for the population, x and y of two...Ch. 3.P1 - Assume that all the rate constants in , are...Ch. 3.P1 - Estimating Eigenvalues and Eigenvectors of from...Ch. 3.P1 - Computing the Entries of from Its Eigenvalues and...Ch. 3.P1 - Given estimates Kij of the entries of K and...Ch. 3.P1 - Table 3.P.1 lists drug concentration measurements...Ch. 3.P2 - If represents the amount of drug (milligrams) in...Ch. 3.P2 - Prob. 2PCh. 3.P2 - Assuming that and , use the parameter values...Ch. 3.P2 - If a dosage is missed, explain through the...Ch. 3.P2 - Suppose the drug can be packaged in a...

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