Elementary Differential Equations
10th Edition
ISBN: 9780470458327
Author: William E. Boyce, Richard C. DiPrima
Publisher: Wiley, John & Sons, Incorporated
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Chapter 2.4, Problem 33P
To determine
To solve: The initial value problem of function
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5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
Chapter 2 Solutions
Elementary Differential Equations
Ch. 2.1 - In each of Problems 1 through 12:
Draw a direction...Ch. 2.1 - In each of Problems 1 through 12:
Draw a direction...Ch. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.1 - Prob. 5PCh. 2.1 - Prob. 6PCh. 2.1 - In each of Problems 1 through 12:
Draw a direction...Ch. 2.1 - Prob. 8PCh. 2.1 - Prob. 9PCh. 2.1 - Prob. 10P
Ch. 2.1 - In each of Problems 1 through 12:
Draw a direction...Ch. 2.1 - In each of Problems 1 through 12:
Draw a direction...Ch. 2.1 - Prob. 13PCh. 2.1 - Prob. 14PCh. 2.1 - In each of Problems 13 through 20, find the...Ch. 2.1 - Prob. 16PCh. 2.1 - Prob. 17PCh. 2.1 - Prob. 18PCh. 2.1 - In each of Problems 13 through 20, find the...Ch. 2.1 - Prob. 20PCh. 2.1 - In each of Problems 21 through 23:
Draw a...Ch. 2.1 - In each of Problems 21 through 23:
Draw a...Ch. 2.1 - In each of Problems 21 through 23:
Draw a...Ch. 2.1 - Prob. 24PCh. 2.1 - Prob. 25PCh. 2.1 - Prob. 26PCh. 2.1 - Prob. 27PCh. 2.1 - Prob. 28PCh. 2.1 - Consider the initial value problem
Find the...Ch. 2.1 - Prob. 30PCh. 2.1 - Prob. 31PCh. 2.1 - Show that all solutions of 2y′ + ty = 2 [Eq. (41)...Ch. 2.1 - Show that if a and λ are positive constants, and b...Ch. 2.1 - Prob. 34PCh. 2.1 - Prob. 35PCh. 2.1 - Prob. 36PCh. 2.1 - Prob. 37PCh. 2.1 - Prob. 38PCh. 2.1 - Prob. 39PCh. 2.1 - Prob. 40PCh. 2.1 - Prob. 41PCh. 2.1 - Prob. 42PCh. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 1 through 8, solve the given...Ch. 2.2 - In each of Problems 9 through 20:
Find the...Ch. 2.2 - Prob. 10PCh. 2.2 - In each of Problems 9 through 20:
Find the...Ch. 2.2 - Prob. 12PCh. 2.2 - Prob. 13PCh. 2.2 - Prob. 14PCh. 2.2 - In each of Problems 9 through 20:
Find the...Ch. 2.2 - Prob. 16PCh. 2.2 - In each of Problems 9 through 20:
Find the...Ch. 2.2 - In each of Problems 9 through 20:
Find the...Ch. 2.2 - Prob. 19PCh. 2.2 - Prob. 20PCh. 2.2 - Prob. 21PCh. 2.2 - Prob. 22PCh. 2.2 - Prob. 23PCh. 2.2 - Solve the initial value problem
y′ = (2 − ex)/(3 +...Ch. 2.2 - Prob. 25PCh. 2.2 - Prob. 26PCh. 2.2 - Prob. 27PCh. 2.2 - Prob. 28PCh. 2.2 - Prob. 29PCh. 2.2 - Prob. 30PCh. 2.2 - Prob. 31PCh. 2.2 - The method outlined in Problem 30 can be used for...Ch. 2.2 - Prob. 33PCh. 2.2 - Prob. 34PCh. 2.2 - The method outlined in Problem 30 can be used for...Ch. 2.2 - The method outlined in Problem 30 can be used for...Ch. 2.2 - The method outlined in Problem 30 can be used for...Ch. 2.2 - The method outlined in Problem 30 can be used for...Ch. 2.3 - Consider a tank used in certain hydrodynamic...Ch. 2.3 - A tank initially contains 120 L of pure water. A...Ch. 2.3 - A tank originally contains 100 gal of fresh water....Ch. 2.3 - A tank with a capacity of 500 gal originally...Ch. 2.3 - Prob. 5PCh. 2.3 - Prob. 6PCh. 2.3 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Prob. 9PCh. 2.3 - A home buyer can afford to spend no more than...Ch. 2.3 - A home buyer wishes to borrow $250,000 at an...Ch. 2.3 - A recent college graduate borrows $150,000 at an...Ch. 2.3 - An important tool in archeological research is...Ch. 2.3 - Suppose that a certain population has a growth...Ch. 2.3 - Suppose that a certain population satisfies the...Ch. 2.3 - Newton’s law of cooling states that the...Ch. 2.3 - Prob. 17PCh. 2.3 - Prob. 18PCh. 2.3 - Prob. 19PCh. 2.3 - A ball with mass 0.15 kg is thrown upward with...Ch. 2.3 - Assume that the conditions are as in Problem 20...Ch. 2.3 - Prob. 22PCh. 2.3 - Prob. 23PCh. 2.3 - Prob. 24PCh. 2.3 - Prob. 25PCh. 2.3 - Prob. 26PCh. 2.3 - Prob. 27PCh. 2.3 - Prob. 28PCh. 2.3 - Prob. 29PCh. 2.3 - Prob. 30PCh. 2.3 - Prob. 31PCh. 2.3 - Prob. 32PCh. 2.4 - Prob. 1PCh. 2.4 - Prob. 2PCh. 2.4 - Prob. 3PCh. 2.4 - Prob. 4PCh. 2.4 - Prob. 5PCh. 2.4 - Prob. 6PCh. 2.4 - Prob. 7PCh. 2.4 - Prob. 8PCh. 2.4 - Prob. 9PCh. 2.4 - Prob. 10PCh. 2.4 - Prob. 11PCh. 2.4 - Prob. 12PCh. 2.4 - Prob. 13PCh. 2.4 - Prob. 14PCh. 2.4 - Prob. 15PCh. 2.4 - Prob. 16PCh. 2.4 - Prob. 17PCh. 2.4 - Prob. 18PCh. 2.4 - Prob. 19PCh. 2.4 - Prob. 20PCh. 2.4 - Prob. 21PCh. 2.4 - Prob. 22PCh. 2.4 - Prob. 23PCh. 2.4 - Prob. 24PCh. 2.4 - Prob. 25PCh. 2.4 - Prob. 26PCh. 2.4 - Prob. 27PCh. 2.4 - Prob. 28PCh. 2.4 - Prob. 29PCh. 2.4 - Prob. 30PCh. 2.4 - Prob. 31PCh. 2.4 - Prob. 32PCh. 2.4 - Prob. 33PCh. 2.5 - Prob. 1PCh. 2.5 - Prob. 2PCh. 2.5 - Prob. 3PCh. 2.5 - Prob. 4PCh. 2.5 - Prob. 5PCh. 2.5 - Prob. 6PCh. 2.5 - Prob. 7PCh. 2.5 - Prob. 8PCh. 2.5 - Prob. 9PCh. 2.5 - Prob. 10PCh. 2.5 - Prob. 11PCh. 2.5 - Prob. 12PCh. 2.5 - Prob. 13PCh. 2.5 - Prob. 14PCh. 2.5 - Prob. 15PCh. 2.5 - Prob. 16PCh. 2.5 - Prob. 17PCh. 2.5 - Prob. 18PCh. 2.5 - Prob. 19PCh. 2.5 - Prob. 20PCh. 2.5 - Prob. 21PCh. 2.5 - Prob. 22PCh. 2.5 - Prob. 23PCh. 2.5 - Prob. 24PCh. 2.5 - Prob. 25PCh. 2.5 - Prob. 26PCh. 2.5 - Prob. 27PCh. 2.5 - Prob. 28PCh. 2.6 - Determine whether each of the equations in...Ch. 2.6 - Prob. 2PCh. 2.6 - Prob. 3PCh. 2.6 - Determine whether each of the equations in...Ch. 2.6 - Prob. 5PCh. 2.6 - Prob. 6PCh. 2.6 - Prob. 7PCh. 2.6 - Prob. 8PCh. 2.6 - Prob. 9PCh. 2.6 - Prob. 10PCh. 2.6 - Determine whether each of the equations in...Ch. 2.6 - Determine whether each of the equations in...Ch. 2.6 - Prob. 13PCh. 2.6 - Prob. 14PCh. 2.6 - Prob. 15PCh. 2.6 - Prob. 16PCh. 2.6 - Prob. 17PCh. 2.6 - Prob. 18PCh. 2.6 - Prob. 19PCh. 2.6 - Prob. 20PCh. 2.6 - Prob. 21PCh. 2.6 - Prob. 22PCh. 2.6 - Prob. 23PCh. 2.6 - Show that if (Nx – My)/(xM – yN) = R, where R...Ch. 2.6 - In each of Problems 25 through 31, find an...Ch. 2.6 - In each of Problems 25 through 31, find an...Ch. 2.6 - Prob. 27PCh. 2.6 - Prob. 28PCh. 2.6 - Prob. 29PCh. 2.6 - In each of Problems 25 through 31, find an...Ch. 2.6 - In each of Problems 25 through 31, find an...Ch. 2.6 - Prob. 32PCh. 2.7 - In each of Problems 1 through 4:
Find approximate...Ch. 2.7 - Prob. 2PCh. 2.7 - In each of Problems 1 through 4:
Find approximate...Ch. 2.7 - Prob. 4PCh. 2.7 - In each of Problems 5 through 10, draw a direction...Ch. 2.7 - Prob. 6PCh. 2.7 - Prob. 7PCh. 2.7 - Prob. 8PCh. 2.7 - Prob. 9PCh. 2.7 - Prob. 10PCh. 2.7 - Prob. 11PCh. 2.7 - Prob. 12PCh. 2.7 - Prob. 13PCh. 2.7 - Prob. 14PCh. 2.7 - Prob. 15PCh. 2.7 - Prob. 16PCh. 2.7 - Prob. 17PCh. 2.7 - Prob. 18PCh. 2.7 - Prob. 19PCh. 2.7 - Convergence of Euler’s Method. It can be shown...Ch. 2.7 - Prob. 21PCh. 2.7 - Prob. 22PCh. 2.7 - Prob. 23PCh. 2.8 - Prob. 1PCh. 2.8 - Prob. 2PCh. 2.8 - Prob. 3PCh. 2.8 - Prob. 4PCh. 2.8 - Prob. 5PCh. 2.8 - Prob. 6PCh. 2.8 - Prob. 7PCh. 2.8 - Prob. 8PCh. 2.8 - Prob. 9PCh. 2.8 - Prob. 10PCh. 2.8 - Prob. 11PCh. 2.8 - Prob. 12PCh. 2.8 - Prob. 13PCh. 2.8 - Prob. 14PCh. 2.8 - Prob. 15PCh. 2.8 - Prob. 16PCh. 2.8 - Prob. 17PCh. 2.8 - Prob. 18PCh. 2.8 - Prob. 19PCh. 2.9 - Prob. 1PCh. 2.9 - Prob. 2PCh. 2.9 - Prob. 3PCh. 2.9 - Prob. 4PCh. 2.9 - Prob. 5PCh. 2.9 - Prob. 6PCh. 2.9 - Find the effective annual yield of a bank account...Ch. 2.9 - An investor deposits $1000 in an account paying...Ch. 2.9 - A certain college graduate borrows $8000 to buy a...Ch. 2.9 - Prob. 10PCh. 2.9 - Prob. 11PCh. 2.9 - Prob. 12PCh. 2.9 - Prob. 13PCh. 2.9 - Prob. 14PCh. 2 - Prob. 1MPCh. 2 - Prob. 2MPCh. 2 - In each of Problems 1 through 32, solve the given...Ch. 2 - Prob. 4MPCh. 2 - Prob. 5MPCh. 2 - Prob. 6MPCh. 2 - Prob. 7MPCh. 2 - Prob. 8MPCh. 2 - Prob. 9MPCh. 2 - Prob. 10MPCh. 2 - Prob. 11MPCh. 2 - Prob. 12MPCh. 2 - Prob. 13MPCh. 2 - Prob. 14MPCh. 2 - Prob. 15MPCh. 2 - Prob. 16MPCh. 2 - Prob. 17MPCh. 2 - Prob. 18MPCh. 2 - Prob. 19MPCh. 2 - Prob. 20MPCh. 2 - Prob. 21MPCh. 2 - Prob. 22MPCh. 2 - Prob. 23MPCh. 2 - Prob. 24MPCh. 2 - Prob. 25MPCh. 2 - Prob. 26MPCh. 2 - Prob. 27MPCh. 2 - Prob. 28MPCh. 2 - Prob. 29MPCh. 2 - Prob. 30MPCh. 2 - Prob. 31MPCh. 2 - Prob. 32MPCh. 2 - Prob. 33MPCh. 2 - Prob. 34MPCh. 2 - Prob. 35MPCh. 2 - Prob. 36MPCh. 2 - Prob. 37MPCh. 2 - Prob. 38MPCh. 2 - Prob. 39MPCh. 2 - Prob. 40MPCh. 2 - Prob. 41MPCh. 2 - Prob. 42MPCh. 2 - Prob. 43MPCh. 2 - Prob. 44MPCh. 2 - Prob. 45MPCh. 2 - Prob. 46MPCh. 2 - Prob. 47MPCh. 2 - Prob. 48MPCh. 2 - Prob. 49MPCh. 2 - Prob. 50MPCh. 2 - Prob. 51MP
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