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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the eigenvalues and eigenfunctions. Verify orthogonality. Start by writing the ODE in the form (1), using Prob. 6. Show details of your work.y" + 8y' + (λ + 16)y = 0, y(0) = 0, y(π) = 0Data
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.
Find from (4a) and formula 3 of Table I in Sec. 11.10. 2 3 + 5 7 8 00 9 10 11 12 f(x) [1 if 0 0) if 0 0) (a > 0) sin x X Jo(ax) (a> 0) T -w/2 sin aw W 2a COS - 1/(40) |E|N 2 n! # (a + wx+1 Re(a +
Find the Fourier series of as given over one period and sketch and partial sums.
Using (8), prove that the series has the indicated sum. Compute the first few partial sums to see that the convergence is rapid.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin
Graph the integrals in Prob. 7, 9, and 11 as functions of x. Graph approximations obtained by replacing ∞ with finite upper limits of your choice. Compare the quality of the approximations. Write a
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.f
Find the steady-state oscillations of y" + cy' + y = r(t) with c > 0 and r(t) as given. Note that the spring constant is k = 1. Show the details. In Probs. 14–16 sketch r(t).
Use Example 2 and R = 1, so that you get the seriesWith the zeros α0,1, α0,2 · · · from your CAS (see also Table A1 in App. 5).(a) Graph the terms J0(α0,1x), · · ·, J0(α0,10x) for 0 ≤ x
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.
Write a short report on ways of obtaining these transforms, with illustrations by examples of your own.
What function do the series of the cosine terms and the series of the sine terms in the Fourier series of ex (-5 < x < 5)represent?
Using (8), prove that the series has the indicated sum. Compute the first few partial sums to see that the convergence is rapid.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.f
Plot Si(u) for positive u. Does the sequence of the maximum and minimum values give the impression that it converges and has the limit π/2? Investigate the Gibbs phenomenon graphically.
Find the Fourier series of as given over one period and sketch and partial sums.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.
Find the steady-state current I(t) in the RLC-circuit in Fig. 275, where R = 10 Ω, L = 1 H, C = 10-1 F and with E(t) V as follows and periodic with period 2π. Graph or sketch the first four partial
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.
Find a Fourier series from which you can conclude that
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x. 0
What could give you the idea to solve Prob. 11 by using the solution of Prob. 9 and formula (9) in the text? Would this work?Data from Prob. 9Find the Fourier transform of (without using Table III in
What function and series do you obtain in Prob. 16 by (term wise) differentiation? Data from Prob. 16 Find the Fourier series of as given over one period and sketch and partial sums.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.
Find the steady-state current I (t) in the RLC-circuit in Fig. 275, where R = 10 Ω, L = 1 H, C = 10-1 F and with E(t) V as follows and periodic with period 2π. Graph or sketch the first four
Show that the familiar identities cos3 x = 3/4 cos x + 1/4 cos 3x and sin3 x = 3/4 sin x -1/4 sin 3x can be interpreted as Fourier series expansions. Develop cos4 x.
Find the half-range expansions of f(x) = x (0 < x < 1).
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.
Represent f (x) as an integral (11).
Find the Fourier series of as given over one period and sketch and partial sums.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.
Represent f (x) as an integral (11).
(a) Write a program for obtaining partial sums of a Fourier series (5).(b) Apply the program to Probs. 8–11, graphing the first few partial sums of each of the four series on common axes. Choose
Solve, y" + ω2y = r(t), where |ω| ≠ 0, 1, 2, · · ·, r(t) is 2π-periodic and
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.
Find the transform (the frequency spectrum) of a general signal of two values [f1 f2]T.
Solve, y" + ω2y = r(t), where |ω| ≠ 0, 1, 2, · · ·, r(t) is 2π-periodic and
Recreate the given signal in Prob. 21 from the frequency spectrum obtained.Data from Prob. 21Find the transform (the frequency spectrum) of a general signal of two values [f1 f2]T.
Find (a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. 4
Compute the minimum square error for f(x) = x/π (-π < x < π) and trigonometric polynomials of degree N = 1, · · ·, 5.
Verify the last statement in Theorem 2 for the discontinuities of f(x) in Prob. 21. Data from Prob. 21 Find the Fourier series of the given function f(x), which is assumed to have the period 2π.
Show that for a signal of eight sample values, w = e-i/4 = (1 - i)/√2. Check by squaring.
Find(a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details.
Same task as in Prob. 23, for f(x) = |x| /π (-π < x π). Why is E* now much smaller (by a factor 100, approximately!)?Data from Prob. 23Compute the minimum square error for f(x) = x/π (-π <
The order seems to be 1/n if f is discontinous, and 1/n2 if f is continuous but f' = df/dx is discontinuous, 1/n3 if f and f' are continuous but f" is discontinuous, etc. Try to verify this for
Calculate the inverse of the 8 × 8 Fourier matrix. Transform a general sample of eight values and transform it back to the given data.
Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.f(x) = x + 1 if 0 <
Find(a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. KIN NIA
Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.f(x) = x if 0 < x
Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena.f(x) = kx if α < x
Find (a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details.f (x) = sin x (0 < x < π)
How does the frequency of the eigenfunctions of the rectangular membrane change(a) If we double the tension?(b) If we take a membrane of half the density of the original one?(c) If we double the
How does the frequency of the fundamental mode of the vibrating string depend on the length of the string? On the mass per unit length? What happens if we double the tension? Why is a contrabass
For what kinds of problems will modeling lead to an ODE? To a PDE?
Show that c is the speed of each of the two waves given by (4).
(a) Graph the basic Fig. 299.(b) In (a) apply animation to “see” the heat flow in terms of the decrease of temperature. (c) Graph u(x, t) with c = 1 as a surface over a rectangle of the form -α
How does the rate of decay of (8) with fixed n depend on the specific heat, the density, and the thermal conductivity of the material?
Prove it for second-order PDEs in two and three independent variables. Prove it by substitution.
Sketch a figure similar to Fig. 317 when c = 1 and f(x) is “triangular,” say, f(x) = x if 0 (t = 0) L (t = 2) (t = 4n) L (t = 6) L 2
Mention some of the basic physical principles or laws that will give a PDE in modeling.
If the first eigenfunction (8) of the bar decreases to half its value within 20 sec, what is the value of the diffusivity?
Radial symmetry reduces (5) to ∇2u = urr + ur/r. Derive this directly from ∇2u = uxx + uyy. Show that the only solution of ∇2u = 0 depending only on r = √x2 + y2 is u = α ln r
Determine and sketch the nodal lines of the square membrane for m = 1, 2, 3, 4 and n = 1, 2, 3, 4.
Write down the derivation in this section for length L = π to see the very substantial simplification of formulas in this case that may show ideas more clearly.
How does the speed of the wave in Example 1 of the text depend on the tension and on the mass of the string?
State three or four of the most important PDEs and their main applications.
If a steel wire 2 m in length weighs 0.9 nt (about 0.20 lb) and is stretched by a tensile force of 300 nt (about 67.4 lb), what is the corresponding speed of transverse waves?
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref(x) = 1/(1 + x2).
Sketch or graph and compare the first three eigenfunctions (8) with Bn = 1, c = 1, and L = π for t = 0, 0.1, 0.2, · · · ,1.0.
Show that (5) can be written ∇2u = (rur)r/r + uθθ/r2, a form that is often practical.
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Wave Equation (1) with suitable cu = cos 4t sin 2x
Write a program for graphing un, with L = π and c2 of your choice similarly as in Fig. 287. Apply the program to u2, u3, u4. Also graph these solutions as surfaces over the xt-plane. Explain the
What is “separating variables” in a PDE? When did we apply it twice in succession?
What are the frequencies of the eigenfunctions in Prob. 3?Data from Prob. 3If a steel wire 2 m in length weighs 0.9 nt (about 0.20 lb) and is stretched by a tensile force of 300 nt (about 67.4 lb),
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref (x) = e-|x|
Find the eigenvalues. Find the corresponding eigenvectors. Use the given λ or factor in Probs. 11 and 15.
Find the growth rate in the Leslie model (see Example 3) with the matrix as given. Show the details.
Find the eigenvalues. Find the eigenvectors.
What is diagonalization? Transformation to principal axes?
Is the matrix A Hermitian or skew-Hermitian? Find x̅TAx. Show the details.
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail.
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.
Find the eigenvalues. Find the corresponding eigenvectors.
Find the limit state of the Markov process modeled by the given matrix. Show the details.
State the definitions and main properties of the three classes of real matrices and of complex matrices that we have discussed.
When can we expect orthogonal eigenvectors?
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail.
Find further 2 × 2 and 3 × 3 matrices without eigenbasis.
Find the eigenvalues. Find the corresponding eigenvectors.
Find the limit state of the Markov process modeled by the given matrix. Show the details.
What is an eigenbasis? When does it exist? Why is it important?
What is algebraic multiplicity of an eigenvalue? Defect?
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors.
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail.
Verify this for A and A = P-1AP. If y is an eigenvector of P, show that x = Py are eigenvectors of A. Show the details of your work.
Find the eigenvalues. Find the corresponding eigenvectors.
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details.
Does a 5 × 5 matrix always have a real eigenvalue?
Verify this for A and A = P-1AP. If y is an eigenvector of P, show that x = Py are eigenvectors of A. Show the details of your work.

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