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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
In Problems 42 through 50, use a calculator or computer system to calculate the eigenvalues and eigenvectors in order to find a general solution of the linear system x' = Ax with the given
The coefficient matrix A of the 4 x 4 systemhas eigenvalues λ1 = -3, λ2 = -6, λ3 = 10, and λ4 = 15. Find the particular solution of this system that satisfies the initial conditions x₁ = 4x1 +
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 4x1 + 2x2, x'2 = 2x1 + 4x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 18 x1 = x1 + 2x2 + 2x3, x2 = 2x1 +7x2 + x3, x3 =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 20 5x1 + x2 + 3x3, x = x + 7x + x3, x3 = 3x1 + x2
In Problems 21 through 24, show that the matrix A is nilpotent and then use this fact to find (as in Example 3) the matrix exponential eAt. A = 1 -1 1 -1
In Problems 21 through 24, show that the matrix A is nilpotent and then use this fact to find (as in Example 3) the matrix exponential eAt. A = 1 ㅜㅜㅇ 1
In Problems 21 through 24, show that the matrix A is nilpotent and then use this fact to find (as in Example 3) the matrix exponential eAt. A = 3 0 5 0 3 0 -3 7 -3
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 9x1 + 2x2, x'2 = 2x1 + 6x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 13x1 + 4x2, x'2 = 4x1 + 7x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = m2 = 1; k1 = 0, k2 = 2, k3 = 0 (no walls)
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = m2 = 1; k1 = 1, k2 = 4, k3 = 1
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = 1 , m2 = 2; k1 = 1, k2 = k3 = 2
Suppose thatShow that A2n = I and that A2n+1 = A if n is a positive integer. Conclude thatand apply this fact to find a general solution of x' = Ax. Verify that it is equivalent to the general
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = m2 = 1 ; k1 = 1, k2 = 2, k3 = 1
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = m2 = 1; k1 = 2, k2 = 1, k3 = 2
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = m2 = 1; k1 = 4, k2 = 6, k3 = 4
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. A = 34 03
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.The mass-and-spring system of Problem 3, with F1(t) Ξ 0, F2(t) = 120 cos
Suppose thatShow that eAt = I cos 2t + 1/2 Asin 2t . Apply this fact to find a general solution of x' = Ax, and verify that it is equivalent to the solution found by the eigenvalue method. -[- A
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. A = 123 01 4 00 1
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. A = 2 0 3 4 1 3 001
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.The mass-and-spring system of Problem 7, with F1(t) = 30 cost, F2(t) = 60
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. 5 A = 0 L 0 20 10 0 30 20 5
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. A = 13 3 3 3 01 3 3 00 23 0002
Apply Theorem 3 to calculate the matrix exponential eAt for each of the matrices in Problems 35 through 40. A = 244 4 420 0 4 00 2 00 4443 4 003
In each of Problems 41 through 46, use the spectral decomposition methods of this section to find a fundamental matrix solution x(t) = eAt for the linear system x' = Ax given in the problem.Problem
In each of Problems 41 through 46, use the spectral decomposition methods of this section to find a fundamental matrix solution x(t) = eAt for the linear system x' = Ax given in the problem.Problem
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 14 x = 3x14x2, x = 4x1+3x2
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 6 x = 9x1 +5x2, x = -6x1-2x2; x1 (0)=1, x2 (0) = 0
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. x(t) = Φ(t)Φ(0) xo. = (8)
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 10x1 - 6x2, x'2 = 12x1 - 7x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 6x1 - 10x2, x'2 = 2x1 - 3x2Data in Problem 9 through 20 9. x = 5x14x2, x = 2x1-x2 x = 4x14x2 10. x =
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problems 15 x = 7x5x2, x2 = 4x1+3x2
Use projection matrices to find a fundamental matrix solution x(t) = eAt of each of the linear systems x' = Ax given in Problems 1 through 20.Problem 16 x = -50x +20x2, x = 100x60x2
In Problems 21 through 24, show that the matrix A is nilpotent and then use this fact to find (as in Example 3) the matrix exponential eAt. ¹ = [₁ A: V 6 4 9- 6-
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problemIn each problem we provide the matrix exponential
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
Each coefficient matrix A in Problems 25 through 30 is the sum of a nilpotent matrix and a multiple of the identity matrix. Use this fact (as in Example 6) to solve the given initial value problem.
Apply the translation theorem to find the Laplace transforms of the functions in Problems 1 through 4. f(t) = e-1/2 cos 2 (1 – हेर) gr
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10.f(t) = sinh t
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. FIGURE 10.1.6. (1, 1) t
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x'' + 9x = 1; x(0) = 0 = x' (0)
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 1 s(s + 4)
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. (1, 1) (2, 1) FIGURE 10.1.7.
Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = s+2 s + 4s +5
Find the inverse Laplace transform f (t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = s(1-e-2s) s² + π² 2
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x" + 4x' + 3x = 1; x (0) = 0 = x'(0)
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 1 ($2 (s + 9)
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. (1, 1) FIGURE 10.1.8.
Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. F(s) = 3s +5 $26s+25
Find the inverse Laplace transform f(t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s) = s(1+e-38) 52 s² +r²
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.x" + 3x' + 2x = t; x (0) = 0, x'(0) = 2
Apply the convolution theorem to find the inverse Laplace transforms of the functions in Problems 7 through 14. F(s) = 1 s (s+k)
Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through 10. (0, 1) (1,0) FIGURE 10.1.9.
Apply the translation theorem to find the inverse Laplace transforms of the functions in Problems 5 through 10. 2s 3 F(s) = 9s2 12s + 20
Find the inverse Laplace transform f(t) of each function given in Problems 1 through 10. Then sketch the graph of f. F(s)= е-лs -e-2πs) 2s(e-s s² +4
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.(x2 - 1)y'' + 4xy' +
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
(a) Deduce from Eqs. (10) and (12) that(b) Use the result of part (a) to verify the formulas in Eq (19) for J1/2(x) and J-1/2(x). 1.3.5 (2n-1) 旦 r(n + 1) = 2²²
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.(x2 + 2)y'' + 4xy' +
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.y'' + xy' + y = 0
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
(a) Suppose that m is a positive integer. Show that(b) Conclude from part (a) and Eq. (13) that r (m + 3) ¹ = 2.5.8...(3m-¹) (3). 3
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.(x2 + 1)y" + 6xy' +
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
Apply Eqs. (19), (26), and (27) to show thatand J3/2(x) = 2 πx3 (sin x - x cos x)
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.(x2 - 3)y" + 2xy' = 0
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
Express J4(x) in terms of J0(x) and J1(x).
In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to
In Problems 1 through 8, determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the
Find general solutions in powers of x of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.(x2 + 3)y" - 7xy' +

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