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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 25 through 30, a city-suburban population transition matrix A (as in Example 2) is given. Find the resulting long-term distribution of a constant total population between the city and its
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. 01 A = [_- ] -1 0
In Problems 1 through 28, determine whether or not the given matrix A is diagonalizable. If it is, find a diagonalizing matrix P and a diagonal matrix D such that P-1 AP = D. 1 1 0 0 01 10 0 0 1
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. =[% A = -6 0
In Problems 25 through 30, a city-suburban population transition matrix A (as in Example 2) is given. Find the resulting long-term distribution of a constant total population between the city and its
In Problems 1 through 28, determine whether or not the given matrix A is diagonalizable. If it is, find a diagonalizing matrix P and a diagonal matrix D such that P-1 AP = D. 1 1
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. A = 0 12 -3 0
In Problems 25 through 30, a city-suburban population transition matrix A (as in Example 2) is given. Find the resulting long-term distribution of a constant total population between the city and its
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. A = 0-12 0 12
In Problems 25 through 30, a city-suburban population transition matrix A (as in Example 2) is given. Find the resulting long-term distribution of a constant total population between the city and its
Problems 31 through 33 deal with a fox-rabbit population as in Examples 3 through 5, except with the transition matrixin place of the one used in the text.If r = 0.16, show that in the long term the
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. ·[₁ A = 0 -6 24 0
Problems 31 through 33 deal with a fox-rabbit population as in Examples 3 through 5, except with the transition matrixin place of the one used in the text.If r = 0.175, show that in the long term the
Find the complex conjugate eigenvalues and corresponding eigenvectors of the matrices given in Problems 27 through 32. = [3% 36 A = 0-4 0 ]
Problems 31 through 33 deal with a fox-rabbit population as in Examples 3 through 5, except with the transition matrixin place of the one used in the text.If r = 0.135, show that in the long term the
Suppose that the 2 x 2 matrix A has eigenvalues λ1 = 1 and λ2 = -1 with eigenvectors v1 = (3, 4) and v2 = (5,7), respectively. Find the matrix A and the powers A99 and A100.
Suppose thatShow that A2n = I and that A2n+1 = A for every positive integer n. 01 =[i]. 1 0 A =
Let A be a 3 x 3 matrix with three distinct eigenvalues. Tell how to construct six different invertible matrices P1, P2,..., P6 and six different diagonal matrices D1, D2,..., D6 such that PiDi
Suppose that the characteristic equation |A - λI| = 0 is written as a polynomial equation [Eq. (5)]. Show that the constant term is c0 = det A. (-1)^2n + Cn-12²¹ +...+c₁d + co = 0, (5)
Suppose thatShow that A4n = I, A4n+1 = A, A4n+2 = -I, and A4n+3 = -A for every positive integer n. A=[- 0 8]. 0
The matrixis not diagonalizable. (Why not?) Write A = I + B. Show that B2 = 0 and thence that A = [1₂] 0
Consider the stochastic matrixwhere 0 < p < 1 and 0 < q < 1. Show that the eigenvalues of A are λ1 = 1 and λ2 = p + q - 1, so that |λ2| < 1. P 1-q A = [1²₂ ¹-9]. 1-
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
In Problems 1 and 2, verify the product law for differentiation, (AB)' = A'B + AB'. A(t) = t 3 1 2t - 1 2¹] and B() = ['172² t 1+t 413 1.
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x'' + 3x' + 7x = t2
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 1 and 2, verify the product law for differentiation, (AB)' = A'B + AB'. et -[₁ -t 8t A (t) = t 0 -1 1² 2 and B(t) = 13 3 2e-t 3t
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x(4) + 6x" - 3x' + x = cos 3t
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = tx - et y + cost, y' = e-t x + t2y - sint
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
Suppose that A is an n x n stochastic matrix—the sum of the elements of each column vector is 1. If v = (1, 1, .... 1), show that AT v = v. Why does it follow that λ = 1 is an eigenvalue of
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = -3y, y' = 3x
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.t2 x" + tx' + (t2 - 1)x = 0
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = 3x - 2y, y' = 2x + y
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.t3 x(3) - 2t2x" + 3tx' + 5x = Int
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = 2x + 4y + 3et, y' = 5x - y - t2
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x(3) = (x')2 + cos x
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
Problems 1 through 7 deal with the mass-and-spring system shown in Fig. 7.5.11 with stiffness matrixand with the given mks values for the masses and spring constants. Find the two natural frequencies
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x" - 5x + 4y = 0, y" + 4x - 5y = 0
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:(a)
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations. = kx (x² + y2)3/2¹ 2 , y" ky (x2 + y2)3/2
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = y + z, y' = z + x z' = x + y
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 8 through 10 the indicated mass-and-spring system is set in motion from rest (x'1 (0) = x'2 (0) = 0) in its equilibrium position (x1(0) = x2(0) = 0) with the given external forces F1(t)
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = 2x - 3y, y' = x + y + 2z, z' = 5y - 7z
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x" + 3x' + 4x - 2y = 0, y" + 2y' - 3x + y = cost
A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the
In Problems 8 through 10 the indicated mass-and-spring system is set in motion from rest (x'1 (0) = x'2 (0) = 0) in its equilibrium position (x1(0) = x2(0) = 0) with the given external forces F1(t)
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = 3x - 4y + z + t, y' = x - 3z + t2, z' = 6y - 7z + t3
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x" = 3x - y + 2z, y" = x + y - 4z, z" = 5x - y - z
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the
In Problems 8 through 10 the indicated mass-and-spring system is set in motion from rest (x'1 (0) = x'2 (0) = 0) in its equilibrium position (x1(0) = x2(0) = 0) with the given external forces F1(t)
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x' = tx - y + et z, y' = 2x + t2 y - z, z' = e-t x + 3ty + t3 z
Consider a mass-and-spring system containing two masses m1 = 1 and m2 = 1 whose displacement functions x(t) and y(t) satisfy the differential equations(a) Describe the two fundamental modes of free
In Problems 1 through 10, transform the given differential equation or system into an equivalent system of first-order differential equations.x" = (1 - y)x, y" = (1 - x)y
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x'1 = x2, x'2 = 2x3, x'3 = 3x4, x'4 = 4x1
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
In Problems 12 and 13, find the natural frequencies of the three-mass system of Fig. 7.5.1, using the given masses and spring constants. For each natural frequency ω, give the ratio a1:a2:a3 of
A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the
In Problems 3 through 12, write the given system in the form x' = P(t)x + f(t).x'1 = x2 + x3 + 1, x'2 = x3 + x4 + t, x'3 = x1 + x4 + t2, x'4 = x1 + x2 + t3
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
Problems 4 through 7 deal with the competition systemin which c1c2 = 9 > 8 = b1b2, so the effect of competition should exceed that of inhibition. Problems 4 through 7 imply that the four critical
(a) Writeby Euler’s formula, expand, and equate real and imaginary parts to derive the identities(b) Use the result of part (a) to find a general solution of cos 3x + i sin 3x = e3ix = (cos x + i
Given a mass m, a dashpot constant c, and a spring constant k, Theorem 2 of Section 5.1 implies that the equationhas a unique solution for t ≧ 0 satisfying given initial conditions x (0) = x0,
Use trigonometric identities to find general solutions of the equations in Problems 44 through 46.y'' + y = x cos3 x
Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x.y'' + 5y' = 0 THEOREM 5 Distinct Real

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