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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Problems 49 and 50 deal with the solution curves of y'' + 3y' + 2y = 0 shown in Figs. 5.1.6 and 5.1.7.Find the highest point on the solution curve with y(0) = 1 and y'(0) = 6 in Fig. 5.1.6.
Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay'' + by' + cy = 0 with constant coefficients. Find such an equation.y(x) = c1 + c2x
Problems 43 through 47 pertain to the solution of differential equations with complex coefficients.Find a general solution of y" = (-2 + 2i√3)y.
The differential equationy'' + (sgn x)y = 0 (25)has the discontinuous coefficient functionShow that Eq. (25) nevertheless has two linearly independent solutions y1(x) and y2(x) defined for all x such
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y'' + 3y' + 2y = 4ex
According to Problem 51 in Section 5.1, the substitution v = ln x (x > 0) transforms the second-order Euler equation ax2y'' + bxy' + cy = 0 to a constant-coefficient homogeneous linear equation.
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y'' - 2y' - 8y = 3e-2x
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y'' - 4y = sinh 2x
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" + 4y = cos 3x
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" + 9y = sin 3x
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x2y'' + xy' + 9y = 0
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" + 9y = 2 sec 3x
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x2y'' + 7xy' + 25y = 0
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" + y = csc2 x
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x3y''' + 6x2y'' + 4xy' = 0
You can verify by substitution that yc = c1x + c2x-1 is a complementary function for the nonhomogeneous second-order equationBut before applying the method of variation of parameters, you must first
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" + 4y = sin2 x
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52–56.x2y'' + xy' = 0
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x3y''' - x2y'' + xy' = 0
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x3y''' + 3x2y'' + xy' = 0
In Problems 47 through 56, use the method of variation of parameters to find a particular solution of the given differential equation.y" - 4y = xex
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x3y''' - 3x2y'' + xy' = 0
In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function yc are given. Apply the method of Problem 57 to find a particular solution of the equation.x2y" -
Make the substitution v = ln x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58.x3y''' + 6x2y'' + 7xy' + y = 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
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 58 through 62, a nonhomogeneous second-order linear equation and a complementary function yc are given. Apply the method of Problem 57 to find a particular solution of the equation.x2y" -
Carry out the solution process indicated in the text to derive the variation of parameters formula in (33) from Eqs. (31) and (32). u₁y₁ + ₂y2 = 0, u₁y₁ + u₂y₂ = f(x) (31)
In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function yc are given. Apply the method of Problem 57 to find a particular solution of the equation.4x2y"
In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function yc are given. Apply the method of Problem 57 to find a particular solution of the equation.x2y" +
In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function yc are given. Apply the method of Problem 57 to find a particular solution of the equation.(x2 -
Apply the variation of parameters formula in (33) to find the particular solution yp (x) = -x cos x of the nonhomogeneous equation y'' + y = 2 sin x. Yp(x) = -y₁ (x) x ) [ ³2 (x)( f(x) - dx +
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 4 -2 1
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 3-2 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. 5 -4 2-1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 5-6 3-4
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 5 -6 3 -4
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. 6 4 -6 -4
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. ヤーヤ 9-9
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 3-1 9- 8
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. 5 -3 2 0
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 4 2 -3 -1
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 4 -3 2 -1
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. 5 3 نیا -4 -2
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 10 -9 6 -5
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 5 -4 3 -2
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. 9-8 6-5
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 6-10 2 -3
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 6-4 3-1
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. 10 -6 12 -7
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 10 -8 6-4
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 13 0 02 20 Lo 02
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. 6-10 2 -3
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 1 0 Lo -2 1 -2 1 0 2
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 7 -6 12 -10
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 8-10 2 -1
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. 11 -15 8- 9
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 1-3 0 2 0 0 1 0 2
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 4 -1 3
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 9-10 2 0
In Problems 1 through 10, a matrix A is given. Use the method of Example 1 to compute A5. 4-3 -1 0 2 0 1 1 2
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. 3-1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 19 -10 21 -10
Find A10 for each matrix A given in Problems 11 through 14. 1 0 6 5 21 -15 -6 926 L21 0 2
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. 5 1 [$ 4] -9
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 13 9 -15 -6
Find A10 for each matrix A given in Problems 11 through 14. 11 -6 20 -11 0 0 -2 -4 1
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. 11 9 -16 -13
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. L 2 2 -2 0 -2 6 0 -1 3
Find A10 for each matrix A given in Problems 11 through 14. 124
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. 130 020 002
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 5 4 -2 0 -4 12 0 -2 6
Find A10 for each matrix A given in Problems 11 through 14. 524 n 199 u -2 -1 -3
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. 222 -2 บร 2-2
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. L 2-2 2 -2 2 222 -2 0 -1 3
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. 3 -3 กา 2-2 0 0 1 1 1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 10 0 -1 3-1 6 0 -2 -6
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. 3 0 -4 -2 1 4 0 0 1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. نیا 3 0 522 1
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. 2 SLA 10 เ พ พ
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. L 1 0 8 0 623 -6 12 -15 -3
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. 6 -5 -3 -2 42 2 2 3
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. L 3 0 6-2 1 0 0 1
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 -2 4 4 L -4 -1 -1 1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 1 -4 10 -15 0 7 0 2 -4
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. 2 0 11 6-15 266 -6 0 2 0
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 4-3 2 -1 0 0 1 1 2
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. 0 1 0 -1 2 0 1 1 -1 -1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 332 6-6 IT 566 9- S
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. 2-2 2 7 7 - 1 L -5 1 0 -1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 0 OWNN 003 200 NNN
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. -2 4 -1 -3 5 -1 -1 1 1
In Problems 1 through 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 1 0 0 0 4 14 0 3 0003 0 0 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. 3-2 1 0 - 1 1 1 1 2
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 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 1 0 0 0 0 1 0 1 1 0 020 002
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. 0 0 1 0 0 พระ
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 26, find the (real) eigenvalues and associated eigenvectors of the given matrix A. Find a basis for each eigenspace of dimension 2 or larger. 4 0 0 6 0 0 0 2 0 -3 0 0 -1 00-5
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 0 0 0 1

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