Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Chapter 9.5, Problem 5E
In Problems 1-8, find the eigenvalues and eigenvectors of the given matrix.
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(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
(6) (8 points) Change the order of integration and evaluate
(z +4ry)drdy .
So S√ ²
0
(10) (16 points) Let R>0. Consider the truncated sphere S given as
x² + y² + (z = √15R)² = R², z ≥0.
where F(x, y, z) = −yi + xj .
(a) (8 points) Consider the vector field
V (x, y, z) = (▼ × F)(x, y, z)
Think of S as a hot-air balloon where the vector field V is the velocity vector
field measuring the hot gasses escaping through the porous surface S. The flux
of V across S gives the volume flow rate of the gasses through S. Calculate
this flux.
Hint: Parametrize the boundary OS. Then use Stokes' Theorem.
(b) (8 points) Calculate the surface area of the balloon. To calculate the surface
area, do the following:
Translate the balloon surface S by the vector (-15)k. The translated
surface, call it S+ is part of the sphere x² + y²+z² = R².
Why do S and S+ have the same area?
⚫ Calculate the area of S+. What is the natural spherical parametrization
of S+?
Chapter 9 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...
Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.2 - In Problems 1-11, find all solutions to the system...Ch. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - In Problems 1-11, find all solutions to the system...Ch. 9.2 - Use the Gauss-Jordan elimination algorithm to show...Ch. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.3 - 9.3 Exercises Let A:=[2135] and B:=[1023] Find a...Ch. 9.3 - Prob. 2ECh. 9.3 - Let A=[2411] and B=[21352] Find a AB. b A2=AA b...Ch. 9.3 - Prob. 4ECh. 9.3 - Let A=[1223], B=[1011] and C=[1121] Find a AB. b...Ch. 9.3 - Prob. 6ECh. 9.3 - 9.3 Exercises a. Show that if u and v are each n1...Ch. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - Prove that if xp satisfy Axp=b, then every...Ch. 9.3 - Let A=[211121112] a. Show that A is singular. b....Ch. 9.3 - In Problems 17-20, find the matrix X1(t) whose...Ch. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.3 - Prob. 22ECh. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - Prob. 27ECh. 9.3 - Prob. 28ECh. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Prob. 33ECh. 9.3 - 9.3 Exercises Find dx/dt for the given matrix...Ch. 9.3 - 9.3 Exercises Verify that the given vector...Ch. 9.3 - Prob. 36ECh. 9.3 - Prob. 37ECh. 9.3 - Prob. 38ECh. 9.3 - Prob. 39ECh. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - Prob. 2ECh. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problem 13-19, determine whether the given...Ch. 9.4 - Let...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - Verify that the vector functions x1=[etet] and...Ch. 9.4 - Verify that the vector functions...Ch. 9.4 - Prove that the operator define by L[x]:=xAx, where...Ch. 9.4 - Let X(t) be a fundamental matrix for the system...Ch. 9.4 - In Problem 29-30, verify that X(t) is a...Ch. 9.4 - Prob. 30ECh. 9.4 - Prob. 31ECh. 9.4 - Abels Formula. If x1,,xn are any n solutions to...Ch. 9.4 - Using Abels formula Problem 32, confirm that the...Ch. 9.4 - Prob. 34ECh. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.4 - Prob. 37ECh. 9.4 - Prob. 38ECh. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 2ECh. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - Consider the system x (t)=Ax(t),t0 with A=[1331]...Ch. 9.5 - Consider the system x(t)=Ax(t),t0 with A=[2112] a....Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - a. Show that the matrix A=[1143] has the repeated...Ch. 9.5 - Prob. 36ECh. 9.5 - Prob. 37ECh. 9.5 - Prob. 38ECh. 9.5 - a. Show that the matrix A=[211121221] has the...Ch. 9.5 - Prob. 40ECh. 9.5 - Prob. 41ECh. 9.5 - Prob. 42ECh. 9.5 - Prob. 43ECh. 9.5 - Prob. 44ECh. 9.5 - Mixing Between Interconnected Tanks. Two tanks,...Ch. 9.5 - Prob. 46ECh. 9.5 - Prob. 48ECh. 9.5 - Stability.A homogeneous system x=Ax with constant...Ch. 9.5 - Prob. 50ECh. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - 9.6 Exercises Find a fundamental matrix for the...Ch. 9.6 - In Problems 13 and 14, find the solution to the...Ch. 9.6 - In Problems 13 and 14, find the solution to the...Ch. 9.6 - Show that x1(t) and x2(t) given by equations 4 and...Ch. 9.6 - Show that x1(t) and x2(t) given by equations 4 and...Ch. 9.6 - In Problems 17 and 18, use the results of Problem...Ch. 9.6 - In Problems 17 and 18, use the results of Problem...Ch. 9.6 - For the coupled mass-spring system governed by...Ch. 9.6 - For the coupled mas-spring system governed by...Ch. 9.6 - Prob. 21ECh. 9.6 - Prob. 22ECh. 9.6 - Stability: In Problem 49 of Exercises 9.5, page...Ch. 9.6 - a. a For Example 1, page 535, verify that...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - Prob. 4ECh. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Prob. 12ECh. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Prob. 14ECh. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Find the solution to the given system that...Ch. 9.7 - 9.7 Exercises In Problems 21 and 22, find the...Ch. 9.7 - Using matrix algebra techniques and method of...Ch. 9.7 - Prob. 24ECh. 9.7 - To find a general solution to the system...Ch. 9.7 - For the system of Problem 25, we found that a...Ch. 9.7 - .Find a general solution of the system...Ch. 9.7 - Prob. 28ECh. 9.7 - Prob. 29ECh. 9.7 - Prob. 30ECh. 9.7 - Prob. 31ECh. 9.7 - Prob. 32ECh. 9.7 - Prob. 33ECh. 9.7 - Prob. 34ECh. 9.8 - In Problems 1-6, a show that the given matrix A...Ch. 9.8 - Prob. 2ECh. 9.8 - Prob. 3ECh. 9.8 - Prob. 4ECh. 9.8 - Prob. 5ECh. 9.8 - Prob. 6ECh. 9.8 - In Problems 7-10, determine eAt by first finding a...Ch. 9.8 - Prob. 8ECh. 9.8 - Prob. 9ECh. 9.8 - In Problems 7-10, determine eAt by first finding a...Ch. 9.8 - In Problems 11 and 12, determine eAt by using...Ch. 9.8 - In Problems 11 and 12, determine eAt by using...Ch. 9.8 - In Problems 17-20, use the generalized...Ch. 9.8 - Prob. 18ECh. 9.8 - Prob. 19ECh. 9.8 - Prob. 20ECh. 9.8 - Prob. 21ECh. 9.8 - Prob. 22ECh. 9.8 - Prob. 23ECh. 9.8 - Prob. 24ECh. 9.8 - Prob. 25ECh. 9.8 - Prob. 26ECh. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 5 and 6, find a fundamental matrix for...Ch. 9.RP - Prob. 6RPCh. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - Prob. 8RPCh. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - In Problems 11 and 12, solve the given initial...Ch. 9.RP - In Problems 11 and 12, solve the given initial...Ch. 9.RP - In Problems 13 and 14, find a general solution for...Ch. 9.RP - In Problems 13 and 14, find a general solution for...Ch. 9.RP - In Problems 15 and 16, find the fundamental matrix...Ch. 9.RP - In Problems 15 and 16, find the fundamental matrix...
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