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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Using the Laplace transform and showing the details, solvey" + 4y = 8t2 if 0 < t < 5 and 0 if t > 5; y(1) = 1 + cos 2, y’ (1) = 4 - 2 sin 2
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work.
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126, assuming i(0) = 0 and: R = 25 Ω, L = 0.1 H, v = 490 e-5t V 0 1
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work.
Solve by the Laplace transform, showing the details and graphing the solution:y" + 4y' + 5y = 50t, y(0) = 5, y' (0) = -5
Using the Laplace transform, find the charge q(t) on the capacitor of capacitance C in Fig. 127 if the capacitor is charged so that its potential is V0 and the switch is closed at t = 0.
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work.
Solve by the Laplace transform, showing the details and graphing the solution:y" - y' - 2y = 12u(t - π) sin t, y(0) = 1, y' (0) = -1
Solve by the Laplace transform, showing the details and graphing the solution:y" + 3y' + 2y = 2u(t - 2), y(0) = 0, y' (0) = 0
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128 with R = 10 Ω and C = 10-2 F, where the current at t = 0 is assumed to be zero, and: v = 0 if t
Solve by the Laplace transform, showing the details and graphing the solution:y'1 = 2y1 - 4y2, y'2 = y1 - 3y2, y1(0) = 3, y2(0) = 0
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 129, assuming zero initial current and charge on the capacitor and: L = 1 H, C = 10-2 F, v = -9900
Solve by the Laplace transform, showing the details and graphing the solution:y'1 = y2 + u(t - π), y'2 = -y1 + u(t - 2π), y1(0) = 1, y2(0) = 0
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 130, assuming zero initial current and charge and: R = 2 Ω, L = 1 H, C = 0.5 F, v(t) = 1 kV if 0 2
Model and solve by the Laplace transform: In Prob. 38, let m1 = m2 = 10 kg, k1 = k3 = 20 kg/sec2, k2 = 40 kg/sec2. Find the solution satisfying the initial conditions y1(0) = y2(0) = 0, y'1(0) = 1
Model and solve by the Laplace transform: Find the current i(t) in the RC-circuit in Fig. 150, where R = 10 Ω, C = 0.1 F, v(t) = 10t V if 0 4, and the initial charge on the capacitor is 0.
Model and solve by the Laplace transform: Find and graph the charge q(t) and the current i(t) in the LC-circuit in Fig. 151, assuming L = 1 H, C = 1 F, v(t) = 1 - e-t if 0 π, and zero initial
Model and solve by the Laplace transform: Find the current i(t) in the RLC-circuit in Fig. 152, where R = 160 Ω, L = 20 H, C = 0.002 F, v(t) = 37 sin 10t V, and current and charge at t = 0 are zero.
Model and solve by the Laplace transform: Set up the model of the network in Fig. 154 and find the solution, assuming that all charges and currents are 0 when the switch is closed at t = 0. Find the
Solve the linear system given explicitly or by its augmented matrix. Show details.4x - 6y = -11-3x + 8y = 10
Why is multiplication of matrices restricted by conditions on the factors?
What properties of matrix multiplication differ from those of the multiplication of numbers?
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
Find three bases of R2.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Let A be a 100 × 100 matrix and B a 100 × 50 matrix. Are the following expressions defined or not? A + B, A2, B2, AB, BA, AAT, BTA, BTB, BBT, BTAB. Give reasons.
Expand a general second-order determinant in four possible ways and show that the results agree.
Solve the linear system given explicitly or by its augmented matrix. Show details. x + y - z = 9 8y + 6z = -6-2x
Are there any linear systems without solutions? With one solution? With more than one solution? Give simple examples.
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
Is the given set, taken with the usual addition and scalar multiplication, a vector space? Give reason. If your answer is yes, find the dimension and a basis.All vectors in R3 satisfying -v1 + 2v2 +
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Solve the linear system given explicitly or by its augmented matrix. Show details.
Let C be 10 × 10 matrix and a a column vector with 10 components. Are the following expressions defined or not? Ca, CTa, CaT, aC, aTC, (CaT)T.
Show that the computation of an nth-order determinant by expansion involves n! multiplications, which if a multiplication takes 10-9 sec would take these times:
Solve the linear system given explicitly or by its augmented matrix. Show details.
Same questions as in Prob. 4 for symmetric matrices.Data from Prob. 4How many different entries can a 4 X 4 skew-symmetric matrix have? An n X n skew-symmetric matrix?
Motivate the definition of matrix multiplication.
If A in Example 2 shows the number of items sold, what is the matrix B of units sold if a unit consists of(a) 5 items(b) 10 items?
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
All polynomials in x of degree 4 or less with non negative coefficients.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Show that det (kA) = Kn det A (not k det A). Give an example.
Explain the use of matrices in linear transformations.
Solve the linear system given explicitly or by its augmented matrix. Show details.
Idempotent matrix defined by A2 = A Can you find four 2 × 2 idempotent matrices?
How can you give the rank of a matrix in terms of row vectors? Of column vectors? Of determinants?
Can you add: A row and a column vector with different numbers of components? With the same number of components? Two row vectors with the same number of components but different numbers of zeros? A
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
All functions y(x) = (αx + b)e-x with any constant α and b.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details. 4y + 3z = 82x - z = 23x + 2y
Nilpotent matrix defined by Bm = 0 for some m. Can you find three 2 × 2 nilpotent matrices?
Showing the details, evaluate: 10.4 4.9 1.5 -1.3
Solve the linear system given explicitly or by its augmented matrix. Show details. -2y - 2z = -83x + 4y - 5z = 13
Can you prove (10a)–(10c) for 3 × 3 matrices? For m × n matrices?
What is the idea of Gauss elimination and back substitution?
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined. 3A, 0.5B, 3A + 0.5B, 3A + 0.5B + C
All 2 x 2 matrices [αjk] with α11 + α22 = 0.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details.
What is the inverse of a matrix? When does it exist? How would you determine it?
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details.
Showing all intermediate results, calculate the following expression or give reasons why they are undefined: AB, ABT, BA, BTA
Find the eigenvalues. Find the corresponding eigenvectors. Use the given λ or factor in Probs. 11 and 15.
Show that the inverse of a skew-symmetric matrix is skew-symmetric.
What do you know about directional derivatives? Their relation to the gradient?
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:a × b, b × a, a • b
Find the work done by a force p acting on a body if the body is displaced along the straight segment A̅B̅ from A to B. Sketch A̅B̅ and p. Show the details.p = [0, 4, 3], A: (4, 5, -1), B: (1, 3,
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:4b × 3c, 12|b × c|, 12|c × b|
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:b × b, (b - c) × (c - b), b • b
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:b, c
Find the resultant in terms of components and its magnitude.p = [-1, 2, -3], q = [1, 1, 1], u = [1, -2, 2]
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:What will happen to the angle in Prob. 24 if we replace c by nc with larger and larger n?Data from prob. 24Let a = [1, 1,
Find the resultant in terms of components and its magnitude.u = [3, 1, -6], v = [0, 2, 5], w = [3, -1, -13]
Same question as in Prob. 34 for two ships moving northeast with speed |vA| = 22 knots and west with speed |vB| = 19 knots.Data from Prob. 34If airplanes A and B are moving southwest with speed |vA|
Forces acting on moving objects (cars, airplanes, ships, etc.) require the engineer to know corresponding tangential and normal accelerations. In Probs. 35–38 find them, along with the velocity and
The use of a CAS may greatly facilitate the investigation of more complicated paths, as they occur in gear transmissions and other constructions. To grasp the idea, using a CAS, graph the path and
Represent in polar form and graph in the complex plane as in Fig. 325. Do these problems very carefully because polar forms will be needed frequently. Show the details. 1 + i
Formulas for hyperbolic functionsShow thatcosh z = cosh x cos y + i sinh x sin ysinh z = sinh x cos y + i cosh x sin y
Prove analyticity of Ln z by means of the Cauchy–Riemann equations in polar form.
Find ez in the form u + iv and |ez| if z equals√2 + 1/2πi
Write in exponential form (6):4 + 3i
Using the definitions, prove:sin z1 cos z2 = 1/2 [sin (z1 + z2) + sin (z1 - z2)]
Find Re and Im ofexp (z3)
Find and graph all roots in the complex plane.4√i
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.u = xy
Apply the program in Prob. 25 to u = ex cos y, v = ex sin y and to an example of your own.Data from Prob. 25Write a program for graphing equipotential lines u = const of a harmonic function u and of
Find and graph all roots in the complex plane.8√1
Find the principal value. Show details.(-1)2-i
Show that if u is harmonic and v is a harmonic conjugate of u, then u is a harmonic conjugate of -v.
Find and graph all roots in the complex plane.5√-1
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.v = -e-2x sin 2y
What is it? Its role? What motivates its name? How can you find it?
Illustrate Prob. 27 by an example.Data from Prob. 27Show that if u is harmonic and v is a harmonic conjugate of u, then u is a harmonic conjugate of -v.
How can you find the answer to Prob. 24 from the answer to Prob. 23?Data from Prob. 24Find the principal value. Show details.(1 - i)1+iData from Prob. 23Find the principal value. Show details.(1 +

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