Questions and Answers of A First Course in Differential Equations

In problem use (5) to find the general solution of the given system. X' = C )x + () cosh t 1 sinh t/
Verify that the vector X is a solution of the given system. 1 0 1 1 -2 0 -1 sin t 0 X; X = -sint - cos t - sin t + cos t X' =
Find the general solution of the given system. 4 0 1 X' = 0 6 0X -4 0 4,
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = 2x - 3y + 1, y(1) = 5; y(1.5)
The RK4 method for solving an initial-value problem over an interval [a, b] results in a finite set of points that are supposed to approximate points on the graph of the exact solution. To expand
Evaluate.(a) Γ(5)(b) Γ(7)(c) Γ(-3/2)(d) Γ(-5/2)
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem find dX/dt. X = 2()e- e2 + 4 -3t
In problem find dX/dt. 5te? 21 X = sin 3t) t sin
Find the eigenvalues and eigenvectors of the given matrix. :)
Find the eigenvalues and eigenvectors of the given matrix. 1 6 0 2 1 1 2.
A square matrix A is said to be a diagonal matrix if all its entries off the main diagonal are zero that is, aij = 0, i ≠ j. The entries aii on the main diagonal may or may not be zero. The
Let A and B be n × n matrices. In general, is(A + B)2 = A2 + 2AB + B2?
If A and B are nonsingular, show that (AB)-1 = B-1A-1.
If A is nonsingular and AB = AC, show that B = C.
Derive formula (3). Find a matrix for which AB = I. Solve for b11, b12, b21, and b22. Then show that BA = I. (b b12 B = b21 b2
If A(t) is a 2 × 2 matrix of differentiable functions and X(t) is a 2 × 1 column matrix of differentiable functions, prove the product rule [A(t)X(t)] = A(1)X'(t) + A'(t)X(t). dt
In problem show that the given matrix has complex eigenvalues. Find the eigenvectors of the matrix. (2 5 -1 4 10 27 2.
In problem show that the given matrix has complex eigenvalues. Find the eigenvectors of the matrix. -1 2) -5 1/
Find the eigenvalues and eigenvectors of the given matrix. 4 -4 -2/ 0 -2,
Find the eigenvalues and eigenvectors of the given matrix. /3 0 2 1/ 14
Find the eigenvalues and eigenvectors of the given matrix. 15 -5 9 0/ -1 (5 -1
Find the eigenvalues and eigenvectors of the given matrix. -8 -1 16 0/
Find the eigenvalues and eigenvectors of the given matrix. :) (2
Find the eigenvalues and eigenvectors of the given matrix. -1 2 8/ -7
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
In problem use Theorem II.3 to find A-1 for the given matrix or show that no inverse exists.Theorem II.3If an n × n matrix A can be transformed into the n × n identity I by a sequence of elementary
LetFind(a) dA/dt(b) dB/dt(c)(d)(e) A(t)B(t)(f) 3t 2 + 1 6t A(t) = and B(t) 12 1/t 4t)
LetFind(a) dA/dt(b)(c) cos A(t) 2t 3t2 - 1
In problem find dX/dt. sin 2t – 4 cos 2t -3 sin 2t + 5 cos 2t) X =
In problem find dX/dt. 5e- 2e 7e/ X =
In problem show that the given matrix is nonsingular for every real value of t. Find A-1(t) using Theorem II.2. (2e' sin t -2e cos t A(t) = e cos t e sin t
In problem show that the given matrix is nonsingular for every real value of t. Find A-1(t) using Theorem II.2.
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem determine whether the given matrix is singular or nonsingular. If it is nonsingular, find A-1 using Theorem II.2.Theorem II.2.Let A be an n × n nonsingular matrix and let Cij =
In problem write the given sum as a single column matrix. -3 -1| 21 – 1+ 8 10 -4 -t 4, -6/
In problem write the given sum as a single column matrix. -7 2) (2 -3 -2' -1 4. 5) -2 3)
In problem write the given sum as a single column matrix. 2 3t t+ (t - 1) -t 31 2 -5t/ 31 4,
In problem write the given sum as a single column matrix. 4 2, -2 + 3 3, (8)
Iffind(a) AT + BT(b) (A + B)T 5 -3 11 and B = -7 A = -4 6/ 2)
Iffind(a) (AB)T(b) BTAT 10 -2 -5) A = 18 and B=
Iffind (a) A + BT(b) 2AT - BT(c) AT(A - B) A = G -2 3 5 7) and B = 2 4
Iffind (a) ATA(b) BT B(c) A + BT 4 8 and B = (2 4 5), -10/ A =
A definition of the gamma function due to Carl Friedrich Gauss that is valid for all real numbers, except x = 0, -1, -2, ..., is given byUse this definition to show that Γ(x + 1) = xΓ(x).
Iffind(a) AB(b) BA(c) (BA)C(d) (AB)C 3 A = (5 -6 7), B = 4. and (1 2 4 C =|0 1 -1 3 2
Use (1) to derive (2) for x > 0.
Iffind(a) BC(b) A(BC)(c) C(BA)(d) A(B + C) 6 3 9, (0 and C = 2 ). B = 4, A %3D -2 2 4.
Use the fact thatto show that Γ(x) is unbounded as x → 0+. I'(x)
Iffind(a) AB(b) BA 4 6 -3 2) A = 5 10 and B 1 -3 18 12
EvaluateLet t = - ln x. 3 x ( In- ax. Jo
Iffind(a) AB(b) BA(c) A2 = AA(d) B2 = BB -3' and B 4 2 A = = -5 3 2,
Use (1) and the fact that Γ(5/3 = 0.89) to evaluate xtedx. I- Jo
Iffind(a) A - B(b) B - A(c) 2(A + B) -2 -1 A = 4 1 and B 7 3, -4 -2/ 3.
Use (1) and the fact that Γ(6/5) = 0.92 to evaluate re'dx. [Hint: Let t = x.]
Iffind(a) A + B(b) B - A(c) 2A + 3B 4 5 -2 A = and B -6 9, 8 - 10,
Answer the question “Why not?” that follows the three sentences after Example 2 on page 366. EXAMPLE 2 Improved Euler's Method Use the improved Euler's method to obtain the approximate value of
A count of the number of evaluations of the function f used in solving the initial-value problem y' = f (x, y), y(x0) = y0 is used as a measure of the computational complexity of a numerical method.
Repeat Problem 19 using the improved Euler’s method, which has global truncation error O(h2). See Problem 5. You might need to keep more than four decimal places to see the effect of reducing the
Repeat Problem 18 for the initial-value problem y' = e-y, y(0) = 0. The analytic solution is y(x) = ln(x + 1). Approximate y(0.5). See Problem 7.Problem 18Consider the initial-value problem y' = 2x -
Repeat Problem 17 for the initial-value problem y' = e-y, y(0) = 0. The analytic solution is y(x) = ln(x + 1). Approximate y(0.5).Problem 17Consider the initial-value problem y' = 2x - 3y + 1, y(1) =
Consider the initial-value problem y' = 2x - 3y + 1, y(1) = 5. The analytic solution is(a) Find a formula involving c and h for the local truncation error in the nth step if the RK4 method is
Repeat Problem 17 using the improved Euler’s method, which has a global truncation error O(h2). See Problem 1. You might need to keep more than four decimal places to see the effect of reducing the
Repeat Problem 16 using the initial-value problem y' = -2y + x, y(0) = 1. The analytic solution isProblem 16Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y(x) =
Consider the initial-value problem y' = 2x - 3y + 1, y(1) = 5. The analytic solution is(a) Find a formula involving c and h for the local truncation error in the nth step if Euler’s method is
Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y(x) = e2x.(a) Approximate y(0.1) using one step and the RK4 method.(b) Find a bound for the local truncation error in
Repeat Problem 15 using the improved Euler’s method. Its global truncation error is O(h2).Problem 15Repeat Problem 13 using the initial-value problem y' = x - 2y, y(0) = 1. The analytic solution
Consider the initial-value problem y'' = x2 + y3, y(1) = 1. See Problem 12 in Exercises 9.1.(a) Compare the results obtained from using the RK4 method over the interval [1, 1.4] with step sizes h =
Repeat Problem 13 using the initial-value problem y' = x - 2y, y(0) = 1. The analytic solution isProblem 13Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y = e2x.(a)
A mathematical model for the area A (in cm2) that a colony of bacteria (B. dendroides) occupies is given by
Repeat Problem 13 using the improved Euler’s method. Its global truncation error is O(h2).Problem 13Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y = e2x.(a)
If air resistance is proportional to the square of the instantaneous velocity, then the velocity v of a mass m dropped from a given height is determined fromLet v(0) = 0, k = 0.125, m = 5 slugs, and
Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y = e2x.(a) Approximate y(0.1) using one step and Euler’s method.(b) Find a bound for the local truncation error in
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = y - y2, y(0) = 0.5; y(0.5)
Although it might not be obvious from the differential equation, its solution could “behave badly” near a point x at which we wish to approximate y(x). Numerical procedures may give widely
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = xy2 - y/x, y(1) = 1; y(1.5) x
Consider the initial-value problem y' = (x + y - 1)2, y(0) = 2. Use the improved Euler’s method with h = 0.1 and h = 0.05 to obtain approximate values of the solution at x = 0.5. At each step
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = xy + √7, y(0) = 1; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = y - y2, y(0) = 0.5; y(0.5)
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = (x - y)2, y(0) = 0.5; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = xy2 - x/y, y(1) = 1; y(1.5)
Use the finite difference method with n = 10 to approximate the solution of the boundary-value problem y'' + 6.55(1 + x)y = 1, y(0) = 0, y(1) = 0.
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'
In problem use the Adams-Bashforth-Moulton method to approximate y(1.0), where y(x) is the solution of the given initial-value problem. First use h = 0.2 and then use h = 0.1. Use the RK4 method to
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = x + y2, y(0) = 0; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = xy + √y, y(0) = 1; y(0.5)
Use Euler’s method with h = 0.1 to approximate x(0.2) and y(0.2), where x(t), y(t) is the solution of the initialvalue problemx' = x + yy' = x - yx(0) = 1, y(0) = 2.
In problem use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then use h = 0.1. Use a numerical solver and h = 0.1 to graph the solution in a neighborhood of t = 0.x'

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