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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 19–22, reduce the given system to echelon form to find a single solution vector u such that the solution space is the set of all scalar multiples of u. x1 + 3x2 + 3x3 + 3x4 = 0 2x1 +
In Problems 19–22, use the method of Example 5 to find the constants A, B, and C in the indicated partial-fraction decompositions. 2x (x + 1)(x + 2)(x + 3) A x+1 + B x+2 + с x+3
In Problems 19–24, use the method of Example 3 to determine whether the given vectors u, v, and w are linearly independent or dependent. If they are linearly dependent, find scalars a, b, and c not
Use the method of Example 9 and the standard integralto derive the general solution y(x) = A cos x + B sin x of the second-order differential equation y'' + y = 0. Thus its solution space has basis
In Problems 25–28, express the vector t as a linear combination of the vectors u, v, and w.t = (0, 0, 19), u = (1,4,3), v = (-1, -2,2), w = (4,4, 1)
In Problems 25–28, express the vector t as a linear combination of the vectors u, v, and w.t = (7,7,7), u = (2, 5, 3), v = (4, 1, -1), w = (1, 1, 5)
Let V be the set of all infinite sequences {xn} = {x1, x2, x3, ....} of real numbers. Let addition of elements of V and multiplication by scalars be defined as follows:and(a) Show that V is a vector
In Problems 29–32, show that the given set V is closed under addition and under multiplication by scalars and is therefore a subspace of R3.V is the set of all (x, y, z) such that x = 0.
In Problems 29–32, show that the given set V is closed under addition and under multiplication by scalars and is therefore a subspace of R3.V is the set of all (x, y, z) such that z = 2x + 3y.
In Problems 29–32, show that the given set V is closed under addition and under multiplication by scalars and is therefore a subspace of R3.V is the set of all (x, y, z) such that 2x + 3y.
In Problems 29–32, show that the given set V is closed under addition and under multiplication by scalars and is therefore a subspace of R3.V is the set of all (x, y, z) such that x + y + z = 0.
In Problems 1–14, a subset W of some n-space Rn is defined by means of a given condition imposed on the typical vector (x1, x2, ...., xn). Apply Theorem 1 to determine whether or not W is a
A sphere in space with center (h, k, l) and radius r has equationFour given points in space suffice to determine the values of h, k, l, and r. In Problems 21 and 22, find the center and radius of the
Use Cramer’s rule to solve the systems in Problems 21–32.11x + 15y = 108x + 11y = 7
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 3x16x22x3 = 1 2x14x2 + x3 = 17 x12x2 - 2x3 = -9
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. = [39] · ₁ 9 1₁ B = [2 |, 87 0 A = 05 0 5 -3
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 3x1 + x2 3x3 = -4 - x1 + x₂ + x3 = x2 1 8 5x1 +
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. A = 4 8 27 4 1 3 , B= 122 103 02 2 -1 10
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. A = 1 2 1 5 1 7 -2 2 , B = 201 3 0 02 0 1
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 2x1 + 5x2 + 12x3 = 6 3x1 + x2 + 5x3 = 12 5x1 + 8x2 +
Use Cramer’s rule to solve the systems in Problems 21–32.5x + 6y = 123x + 4y = 6
Use Cramer’s rule to solve the systems in Problems 21–32.6x + 7y = 38x + 9y = 4
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. A = 1 2 2 -2 1 2 3 7 7 -[i 0 0 1 B = 01 1 101 0 1 0
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. х1 - 4х2 - 3x3 - 3x4 X1 2х1 - 6х2 - 5х3 -
Use Cramer’s rule to solve the systems in Problems 21–32.5x1 + 2x2 - 2x3 = 1x1 + 5x2 - 3x3 = -25x1 - 3x2 + 5x3 = 2
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. A = 653 5 3 342 2 , B= 210 2 -1 3 3 5 0 1105
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 3x16x2 + x3 + 13x4 = 15 3х1 - 6х2 + 3x3 + 21x4 =
Use Cramer’s rule to solve the systems in Problems 21–32.5x1 + 4x2 - 2x3 = 42x1 + 3x3 = 22x1 - x2 + x3 = 1
Under what condition on the constants a, b, and c does the systemhave a unique solution? No solution? Infinitely many solutions? 2x = y + 3z = a x + 2y + z = b 7x + 4y + 9z = c
A diagonal matrix is a square matrix of the form in which every element off the main diagonal is zero. Show that the product AB of two n x n diagonal matrices A and B is again a diagonal matrix.
The formula in Problem 29 can be used to compute A2 without an explicit matrix multiplication. It follows thatwithout an explicit matrix multiplicationand so on. Use this method to compute A2, A3,
The positive integral powers of a square matrix A are defined as follows:Suppose that r and s are positive integers. Prove that Ar As = Ar+s and that (Ar)s = Ars (in close analogy with the laws of
Problems 28 through 30 develop a method of computing powers of a square matrix.Ifthen show thatwhere I denotes the 2 x 2 identity matrix. Thus every 2 x 2 matrix A satisfies the equationwhere det A =
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 3x1 + x2 + x3 + 6x4 = 14 x12x2 + 5x3 - 5x4 = -7 4x1
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 2x1 + 4x2 x3 2x4 + 2x5 = 6 x1 + 3x2 + 2x3-7x4 + 3x5
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. -5-2 2 1 5 L 5 -3 23 -3 1
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. 3 5 -2 3 دیا L -5 2 -4 0 -5
Verify parts (a) and (b) of Theorem 3. THEOREM 3 Algebra of Inverse Matrices If the matrices A and B of the same size are invertible, then (a) A¹ is invertible and (A-1)-¹ = A; (b) If n is a
Use Cramer’s rule to solve the systems in Problems 21–32. 2х1 4х1 - 5x2 + 3x3 = -2x1 + x2 + X3 = Х2 5x3 =-3 3 1
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. 37 2 0 -5 -4 L 2 -1 1 32
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. L -4 4 3 -1 1 0 3 355 -5 -5
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. NOW نیا ترا 3-5
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. -4 พ 15 55 4 -3 -1
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. -3 116
Use Cramer’s rule to solve the systems in Problems 21–32.3x1 - x2 - 5x3 = 34x1 - 4x2 - 3x3 = -4x1 - 5x3 = 2
Apply Theorem 5 to find the inverse A-1 of each matrix A given in Problems 33–40. 4 -3 -5 0 225 ㅗ
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
Use Cramer’s rule to solve the systems in Problems 21–32.x1 + 4x2 + 2x3 = 34x1 + 2x2 + x3 = 12x1 - 2x2 - 5x3 = -3
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
Show that the i th row of the product AB is AiB, where Ai is the i th row of the matrix A. AIA and IB = B (2)
Suppose that A is an invertible matrix and that r and s are negative integers. Verify that Ar As = Ar+s and that (Ar)s = Ars.
Suppose that A, B, and C are invertible matrices of the same size. Show that the product ABC is invertible and that (ABC)-1 = C-1 B-1 A-1.
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
Problems 23 through 26 introduce the idea—developed more fully in the next section—of a multiplicative inverse of a square matrix.LetandFind B so that AB = I = BA as follows: First equate entries
In Problems 23–28, use the method of Example 8 to find a matrix X such that AX = B. ^= A 4 [$ 3]. B = [-1 [₁ 5 - ] 3 -2 -5 5
Use Cramer’s rule to solve the systems in Problems 21–32.17x + 7y = 612x + 5y = 4
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. X1 + 3x2 + 3x3 = 13. 2x1 + 5x2 + 4x3 = 23 2x1 + 7x2
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 3x1 + X2 - 3x3 = 6 2x1 + 7x2 + x3 =-9 2x1 + 5x2 = -5
Use the method of Gauss-Jordan elimination (transforming the augmented matrix into reduced echelon form) to solve Problems 11–20 in Section 3.2. 2x1 + 8x2 + 3x3 = 2 x1 + 3x2 + 2x3 = 5 2x1 + 7x2 +
Let E be the elementary matrix E1 of Example 6. If A is a 2 x 2 matrix, show that EA is the result of multiplying the first row of A by 3.Example 6 Example 6 We obtain some typical elementary
Consider the 3 x 3 matrixFirst verify by direct computation that A2 = 3A. Then conclude that An+1 = 3nA for every positive integer n. A = 2-1 -1 -1 2 -1 -1 2 -1
Consider the 2 x 2 matriceswhere x and y denote the row vectors of B. Then the product AB can be written in the formUse this expression and the properties of determinants to show thatThus the
Problems 31-38 illustrate ways in which the algebra of matrices is not analogous to the algebra of real numbers.This is a continuation of the previous two problems. Find two nonzero 2 x 2 matrices A
Let E be the elementary matrix E3 of Example 6. Show that EA is the result of interchanging the first two rows of the matrix A.Example 6 Example 6 We obtain some typical elementary matrices as
Let E be the elementary matrix E2 of Example 6 and suppose that A is a 3 x 3 matrix. Show that EA is the result upon adding twice the first row of A to its third row.Example 6 Example 6 We obtain
Use the formula of Problem 29 to find a 2 x 2 matrix A such that A ≠ 0 and A ≠ I but such that A2 = A.Problem 29where I denotes the 2 x 2 identity matrix. Thus every 2 x 2 matrix A satisfies the
Find a 2 x 2 matrix A with each element +1 or -1 such that A2 = 0. The formula of Problem 29 may be helpful.Problem 29where I denotes the 2 x 2 identity matrix. Thus every 2 x 2 matrix A satisfies
Use matrix multiplication to show that if x1 and x2 are two solutions of the homogeneous system Ax = 0 and c1 and c2 are real numbers, then c1x1 + c2x2 is also a solution.
Find a 2 x 2 matrix A with each main diagonal element zero such that A2 = -I.
Find a 2 x 2 matrix A with each main diagonal element zero such that A2 = I.
Problems 31-38 illustrate ways in which the algebra of matrices is not analogous to the algebra of real numbers.(a) Suppose that A and B are the matrices of Example 5. Show that (A + B)(A - B) ≠ A2
List all possible reduced row-echelon forms of a 2 x 2 matrix, using asterisks to indicate elements that may be either zero or nonzero.
Find four different 2 x 2 matrices A, with each main diagonal element either +1 or -1, such that A2 = I.
Use Cramer’s rule to solve the systems in Problems 21–32.3x1 + 4x2 - 3x3 = 53x1 - 2x2 + 4x3 = 73x1 + 2x2 - x3 = 3
Each of Problems 43–46 lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column. a21 922 923 a11
Let A = [ahi], B = [bij], and C = [Cjk] be matrices of sizes m x n, n x p, and p x q, respectively. To establish the associative law A(BC) = (AB)C, proceed as follows. By Equation (16) the hjth
Each of Problems 43–46 lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column. kan a12
Problems 61–64 deal with the Vandermonde determinant that will play an important role in Section 3.7.Show by direct computation that V(a, b) = b - a and that V(x1,X2,....Xn) = 1 X1 1 x2
Problems 61–64 deal with the Vandermonde determinantUse the formula in (25) to evaluate the two determinants given next. V(x1,x2,...,xn): = 1 1 X1 x² x2 x2 x2 LE 1 xn x 2 A xn-1 n-1 ... n .n-1 n
Consider the n x n determinantin which each entry on the main diagonal is a 2, each entry on the two adjacent diagonals is a 1, and every other entry is zero.(a) Expand along the first row to show
Figure 3.6.2 shows an acute triangle with angles A, B, and C and opposite sides a, b, and c. By dropping a perpendicular from each vertex to the opposite side, derive the equationsRegarding these as
Each of Problems 43–46 lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column. a₁ b₁ c₁
Suppose that A2 = A. Prove that |A| = 0 or |A| = 1.
Let A = [aij ] be a 3 x 3 matrix. Show that det(AT) = det A by expanding det A along its first row and det(AT) along its first column.
Problems 47 through 49 develop properties of matrix transposes.Suppose that A and B are matrices of the same size. Show that:(a) (AT)T = A;(b) (CA)T = CAT; and(c) (A + B)T = AT + BT.
Each of Problems 43–46 lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column. a11
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use Euler’s method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to four decimal
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge–Kutta method with step sizes h = 0.2, 0.1, 0.05, and 0.025 to approximate to six
Consider the crossbow bolt of Example 3, shot straight upward from the ground (y = 0) at time t = 0 with initial velocity v0 = 49 m/s. Take g = 9.8 m/s2 and ρ = 0.0011 in Eq. (12). Then use Eqs.
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use Euler’s method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to four decimal
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge–Kutta method with step sizes h = 0.2, 0.1, 0.05, and 0.025 to approximate to six
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the improved Euler method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use Euler’s method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to four decimal
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
In each of Problems 23-30, a second-order differential equation and its general solution y(x) are given. Determine the constants A and B so as to find a solution of the differential equation that
A computer with a printer is required for Problems 17 through 24. In these initial value problems, use the Runge–Kutta method with step sizes h = 0.2, 0.1, 0.05, and 0.025 to approximate to six

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