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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Textbook Question
Chapter 2.6, Problem 8P
Exact Equations. In each of Problem
Determine whether the equation is exact. If it is exact, then:
Solve the equation.
Use a computer to draw several integral curves.
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1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show…
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1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show that P - Q is its own inverse.
4. Show that the Frobenius product on n x n-matrices,
(A, B) =
= Tr(B*A),
is an inner product, where B* denotes the Hermitian adjoint of B.
5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen-
vectors (for both A and B), then AB = BA.
Remark: It is also true that if AB = BA, then there exists a common…
Chapter 2 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 2.1 - In each of Problems 1 through 12, solve the given...Ch. 2.1 - In each of Problems through , solve the given...Ch. 2.1 - In each of Problems 1 through 12, solve the given...Ch. 2.1 - In each of Problems 1 through 12, solve the given...Ch. 2.1 - In each of Problems 1 through 12, solve the given...Ch. 2.1 - In each of Problems through , solve the given...Ch. 2.1 - In each of Problems through , solve the given...Ch. 2.1 - In each of Problems 1 through 12, solve the given...Ch. 2.1 - In each of Problems through , solve the given...Ch. 2.1 - In each of Problems through , solve the given...
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(a) Find the...Ch. 2.1 - In each of Problems 13 through 28: (a) Find the...Ch. 2.1 - In each of Problems through :
(a) Find the...Ch. 2.1 - In Problems through , obtain the requested...Ch. 2.1 - In Problems 29 through 36, obtain the requested...Ch. 2.1 - In Problems through , obtain the requested...Ch. 2.1 - In Problems 29 through 36, obtain the requested...Ch. 2.1 - In Problems through , obtain the requested...Ch. 2.1 - In Problems 29 through 36, obtain the requested...Ch. 2.1 - In Problems through , obtain the requested...Ch. 2.1 - In Problems 29 through 36, obtain the requested...Ch. 2.1 - Solve the equation dydx=ay+bcy+d, where a,b,c, and...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12:
Draw a...Ch. 2.2 - In each of Problems 1 through 12:
Draw a...Ch. 2.2 - In each of Problems 1 through 12:
Draw a...Ch. 2.2 - In each of Problems 1 through 12:
Draw a...Ch. 2.2 - In each of Problems 1 through 12:
Draw a...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 1 through 12: Draw a direction...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 13 through 20, find the...Ch. 2.2 - In each of Problems 21 through 23:
Draw a...Ch. 2.2 - In each of Problems 21 through 23:
Draw a...Ch. 2.2 - In each of Problems 21 through 23: Draw a...Ch. 2.2 - In each of Problems 21 through 23:
Draw a...Ch. 2.2 - In each of Problems 24 through 26:
Draw a...Ch. 2.2 - In each of Problems 24 through 26: Draw a...Ch. 2.2 - In each of Problems 24 through 26:
Draw a...Ch. 2.2 - Consider the initial value problem
Find the...Ch. 2.2 - Consider the initial value problem
Find the value...Ch. 2.2 - Consider the initial value problem...Ch. 2.2 - Find the value of y0 for which the solution of the...Ch. 2.2 - Consider the initial value problem
Find the value...Ch. 2.2 - Show that all solutions of [Eq. (36) of the text]...Ch. 2.2 - Show that if andare positive constants, and b is...Ch. 2.2 - In each of Problems 34 through 37, construct a...Ch. 2.2 - In each of Problems 34 through 37, construct a...Ch. 2.2 - In each of Problems 34 through 37, construct a...Ch. 2.2 - In each of Problems 34 through 37, construct a...Ch. 2.2 - Consider the initial value problem...Ch. 2.2 - Variation of Parameters. Consider the following...Ch. 2.2 - In each of Problems 40 through 43 use the method...Ch. 2.2 - In each of Problems 40 through 43 use the method...Ch. 2.2 - In each of Problems 40 through 43 use the method...Ch. 2.2 - In each of Problems 40 through 43 use the method...Ch. 2.3 - Consider a tank used in certain hydrodynamic...Ch. 2.3 - A tank initially contains 200L of pure water. A...Ch. 2.3 - A tank originally contains gal of fresh water....Ch. 2.3 - A tank with a capacity of originally contains of...Ch. 2.3 - A tank contains of water and of salt. Water...Ch. 2.3 - Suppose that a tank containing a certain liquid...Ch. 2.3 - An outdoor swimming pool loses 0.05 of its water...Ch. 2.3 -
Cholesterol is produced by the body for the...Ch. 2.3 - Imagine a medieval world. In this world a Queen...Ch. 2.3 - Suppose an amount is invested at an annual rate...Ch. 2.3 - A young person with no initial capital invests ...Ch. 2.3 - A homebuyer can afford to spend no more than on...Ch. 2.3 - A recent college graduate borrows 100,000 at an...Ch. 2.3 - A Difference Equation. In this problem, we...Ch. 2.3 - An important tool in archaeological research is...Ch. 2.3 - The population of mosquitoes in a certain area...Ch. 2.3 - Suppose that a certain population has growth rate...Ch. 2.3 - Suppose that a certain population satisfies the...Ch. 2.3 - Newtons law of cooling states that the temperature...Ch. 2.3 - Heat transfer from a body to its surrounding by...Ch. 2.3 - Consider a lake of constant volume containing at...Ch. 2.3 - A ball with mass 0.25 kg is thrown upward with...Ch. 2.3 - Assume that conditions are as Problemexcept that...Ch. 2.3 - Assume that conditions are as in Problem 22 except...Ch. 2.3 - A skydiver weighing 180 lb (including equipment)...Ch. 2.3 - A rocket sled having an initial speed of mi/h is...Ch. 2.3 - A body of constant mass is projected vertically...Ch. 2.3 - Prob. 28PCh. 2.3 - Prob. 29PCh. 2.3 - A mass of 0.40 kg is dropped from rest in a medium...Ch. 2.3 - Suppose that a rocket is launched straight up from...Ch. 2.3 - Let and , respectively, be the horizontal and...Ch. 2.3 - A more realistic model (than that in Problem 32)...Ch. 2.3 - Brachistochrone Problem. One of the famous...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - Existence and uniqueness of Solutions. In each of...Ch. 2.4 - In each of Problem through, state where in -...Ch. 2.4 - In each of Problem through, state where in -...Ch. 2.4 - In each of Problem through, state where in -...Ch. 2.4 - In each of Problem 7 through 12, state where in...Ch. 2.4 - In each of Problem through, state where in -...Ch. 2.4 - In each of Problem through, state where in -...Ch. 2.4 - Consider the initial value problem y=y1/3,y(0)=0...Ch. 2.4 -
Verify that both and are solutions of the...Ch. 2.4 - Dependence of Solutions on Initial Conditions. In...Ch. 2.4 - Dependence of Solutions on Initial Conditions. In...Ch. 2.4 - Dependence of Solutions on Initial Conditions. In...Ch. 2.4 - Dependence of Solutions on Initial Conditions. In...Ch. 2.4 - In each of Problem 19 through 22, draw a direction...Ch. 2.4 - In each of Problem 19 through 22, draw a direction...Ch. 2.4 - In each of Problem through, draw a direction...Ch. 2.4 - In each of Problem through, draw a direction...Ch. 2.4 -
Show that is a solution of and that is also a...Ch. 2.4 - Show that if y=(t) is a solution of y+p(t)y=0,...Ch. 2.4 - Let y=y1(t) be a solution of y+p(t)y=0, (i) and...Ch. 2.4 -
Show that the solution (7) of the general...Ch. 2.4 - Discontinuous Coefficients. Linear differential...Ch. 2.4 - Discontinuous Coefficients. Linear differential...Ch. 2.4 - Consider the initial value problem
...Ch. 2.5 - Suppose that a certain population obeys the...Ch. 2.5 - Another equation that has been used to model...Ch. 2.5 - (a) Solve the Gompertz equation subject to the...Ch. 2.5 - A pond forms as water collects in a conical...Ch. 2.5 - Consider a cylindrical water tank of constant...Ch. 2.5 - Epidemics. The use of mathematical methods to...Ch. 2.5 - Epidemics. The use of mathematical methods to...Ch. 2.5 - Epidemics. The use of mathematical methods to...Ch. 2.5 - Chemical Reactions. A second order chemical...Ch. 2.5 - Bifurcation Points. For an equation of the form...Ch. 2.5 - Bifurcation Points. For an equation of the form
...Ch. 2.5 - Bifurcation Points. For an equation of the form...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem 1 through 12:...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem 1 through 12:...Ch. 2.6 - Exact Equations. In each of Problem 1 through 12:...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem 1 through 12:...Ch. 2.6 - Exact Equations. In each of Problem 1 through 12:...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - Exact Equations. In each of Problem through...Ch. 2.6 - In each of Problem and , solve the given initial...Ch. 2.6 - In each of Problem 13 and 14, solve the given...Ch. 2.6 - In each of Problem 15 and 16, find the value of b...Ch. 2.6 - In each of Problem 15 and 16, find the value of b...Ch. 2.6 - Assume that Eq. (6) meets the requirements of...Ch. 2.6 - Show that any separable equation is also exact.
Ch. 2.6 - Integrating Factor. In each of Problem through...Ch. 2.6 - Integrating Factor. In each of Problem through...Ch. 2.6 - Integrating Factor. In each of Problem 19 through...Ch. 2.6 - Integrating Factor. In each of Problem through...Ch. 2.6 - Show that if (NxMy)/M=Q, where Q is function of y...Ch. 2.6 - Show that if , where depends on the quantity ...Ch. 2.6 - In each of Problem 25 through 31: Find an...Ch. 2.6 - In each of Problem through:
Find an integrating...Ch. 2.6 - In each of Problem 25 through 31: Find an...Ch. 2.6 - In each of Problem 25 through 31: Find an...Ch. 2.6 - In each of Problem through:
Find an integrating...Ch. 2.6 - In each of Problem 25 through 31: Find an...Ch. 2.6 - In each of Problem 25 through 31: Find an...Ch. 2.6 - Use the integrating factor (x,y)=[xy(2x+y)]1 to...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - Homogeneous Differential Equations. In each of...Ch. 2.7 - In problem 11 and 12, solve the given initial...Ch. 2.7 - In problem and, solve the given initial value...Ch. 2.7 - In each of Problems 13 through 22: Write the...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems 13 through 22: Write the...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - In each of Problems 13 through 22: Write the...Ch. 2.7 - In each of Problems through:
Write the Bernoulli...Ch. 2.7 - A differential equation of the form...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems ...Ch. 2.7 - Mixed Practice. In each of Problems 24 through 36:...Ch. 2.P1 - Constant Effort Harvesting. At a given level of...Ch. 2.P1 - Constant Yield Harvesting. In this problem, we...Ch. 2.P2 - Derive Eq. (3) from Eqs. (1) and (2) and show that...Ch. 2.P2 - Additional processes due to biotic and abiotic...Ch. 2.P2 - Show that when , the source has an infinite...Ch. 2.P2 - Assume the following values for the parameters;...Ch. 2.P2 - Effects of Partial Source Remediation.
Assume...Ch. 2.P3 - Simulate five sample trajectories of Eq. (1) for...Ch. 2.P3 - Use the difference equation (4) to generate an...Ch. 2.P3 - VarianceReduction by Antithetic Variates. A simple...
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