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mathematics
advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = grad (x3 cos2 (xy)), R as in Prob. 5Data from Prob. 5Evaluate ∫C F(r) • dr
Describe the region of integration and evaluate.
Show that the form under the integral sign is exact in the plane (Probs. 3–4) or in space (Probs. 5–9) and evaluate the integral. Show the details of your work.
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(x2D2 - 4xD + 6I)y = 21x-4
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.y" + y = cos x – sin x
Find the steady-state current in the RLC-circuit in Fig. 61 for the given data. Show the details of your work.R = 12 Ω, L = 0.4 H, C = 1/80 F, E = 220 sin 10t V
Find the steady-state current in the RLC-circuit in Fig. 61 for the given data. Show the details of your work.R = 4 Ω, L = 0.1 H, C = 0.05 F, E = 110 V
This is an RLC-circuit with negligibly small R (analog of an undamped mass–spring system). Find the current when L = 0.5 H, C = 0.005 F, and E = sin t V, assuming zero initial current and charge.
Model the RL-circuit in Fig. 66. Find a general solution when R, L, E are any constants. Graph or sketch solutions when L = 0.25 H, R = 10 Ω, and E = 48 V.
Model the RC-circuit in Fig. 64. Find the current due to a constant E.
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2+ 2I)y = cos √2t + sin √2t
Write a condensed report of 2–3 pages on the most important similarities and differences of free and forced vibrations, with examples of your own. No proofs.
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.8y" - 6y' + y = 6 cosh x, y(0) = 0.2, y' (0) = 0.05
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.y" + 3y = 18x2, y(0) = -3, y' (0) = 0
Find a (real) general solution. State which rule you are using. Show each step of your work.y" + 4y' + 4y = e-x cos x
(a) Find a second-order homogeneous linear ODE for which the given functions are solutions. (b) Show linear independence by the Wronskian. (c) Solve the initial value problem.e-2·5x cos 0.3x,
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.cosh αx, sinh αx
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.x3, x2
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.x, 1/x
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.e-0.4x, e-2.6x
Derive (6*) from (6).
Find a real general solution. Show the details of your work.4x2y" + 5y = 0
Study this transition in terms of graphs of typicalsolutions. (Cf. Fig. 47.)(a) Avoiding unnecessary generality is part of goodmodeling. Show that the initial value problems (A)and (B),(A)(B) the
Show that the ratio oftwo consecutive maximum amplitudes of a dampedoscillation (10) is constant, and the natural logarithmof this ratio called the logarithmic decrement, equals . Find for the
Consider an under damped motion of a body of mass m = 0.5 kg. If the time between two consecutive maxima is 3 sec and the maximum amplitude decreases to 1/2 its initial value after 10 cycles, what is
Show that for (7) to satisfy initial conditions y(0) = y0 and v(0) = v0 we must have c1 = [(1 + α/β)y0 + v0/β]/2 and c2 = [(1 - α/β)y0 - v0/β]/2.
If 1 liter of water (about 1.06 US quart) is vibrating up and down under the influence of gravitation in a U-shaped tube of diameter 2 cm (Fig. 44), what is the frequency? Neglect friction. First
What are the frequencies of vibration of a body of mass m = 5 kg (i) on a spring of modulus k1 = 20 nt/m, (ii) on a spring of modulus k2 = 45 nt/m, (iii) on the two springs in parallel? See Fig. 41.
Apply the given operator to the given functions. Show all steps in detail.(D2 - 4.00D + 3.84I)y = 0
Apply the given operator to the given functions. Show all steps in detail.(D - 2I)(D + 3I); e2x, xe2x, e-3x
Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.4y" - 4y' - 3y = 0
Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.y"+ 1.8y' - 2.08y = 0
Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.y" + 4.5y' = 0
Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.y" + 2πy' + π2y = 0
Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9.y" + 6y' + 8.96y = 0
Write a program for testing linear independence and dependence. Try it out on some of the problems in this and the next problem set and on examples of your own.
Find the curve through the origin in the xy-plane which satisfies y'' = 2y' and whose tangent at the origin has slope 1.
Reduce to first order and solve, showing each step in detail.x2y'' - 5xy' + 9y = 0y1 = x3
Reduce to first order and solve, showing each step in detail.yy'' = 3y'2
What other solution methods did we consider in this chapter?
Explain the basic concepts ordinary and partial differential equations (ODEs, PDEs), order, general and particular solutions, initial value problems (IVPs). Give examples.
Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.y' = 3.2y - 10y2
These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear
These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear
These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear
A thermometer, reading 5°C, is brought into a room whose temperature is 22°C. One minute later the thermometer reading is 12°C. How long does it take until the reading is practically 22°C, say,
Find a general solution. Show the steps of derivation. Check your answer by substitution.y' = sec2 y
Model the motion of a body B on a straight line with velocity as given, y(t) being the distance of B from a point y = 0 at time t. Graph a direction field of the model (the ODE). In the field sketch
Direction fields are very useful because they can give you an impression of all solutions without solving the ODE, which may be difficult or even impossible. To get a feel for the accuracy of the
Half-life. The half-life measures exponential decay. It is the time in which half of the given amount of radioactive substance will disappear. What is the halflife of 22688 Ra (in years) in Example
Find two constant solutions of the ODE in Prob. 13 by inspection.Data from prob 13y' = y - y2y = 1/1 + ce-xy(0) = 0.25
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [ey, ez, ex], S: z = x2 (0 ≤ x ≤ 2, 0 ≤ y ≤ 1)
Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [z, ex,
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [e-z, e-z, cos y, e-z sin y], S: z = y2/2, -1 ≤ x ≤ 1, 0 ≤ y ≤ 1
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [x2, y2, z2], S the surface of x2 + y2 ≤ z2, 0 ≤ z ≤ h
Show that w = ex sin y satisfies Laplace’s equation ∇2w = 0 and, using (12), integrate w(∂w/∂n) counterclockwise around the boundary curve C of the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 5.
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y' + 5.2y = 19.4 sin 2t, y(0) = 0
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.3t + 12
Explain and compare the different roles of the two shifting theorems, using your own formulations and simple examples. Give no proofs.
Make a draft of these four operations from memory. Then compare your draft with the text and write a 2- to 3-page report on these operations and their significance in applications.
State the Laplace transforms of a few simple functions from memory.
Consider a vibrating system of your choice modeled byy" + cy' + ky = δ(t).(a) Using graphs of the solution, describe the effect of continuously decreasing the damping to 0, keeping k constant.(b)
Find:1 * 1
Showing the details of your work, find L(f) if f(t) equals:3t sinh 4t
What are the steps of solving an ODE by the Laplace transform?
Find:1 * sin ωt
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - y' - 6y = 0, y(0) = 11, y' (0) = 28
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.cos πt
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.t -2 (t > 2)
Showing the details of your work, find L(f) if f(t) equals:1/2 te-3t
Find:et * e-t
Find and graph or sketch the solution of the IVP. Show the details.y" + 4y = δ(t - π), y(0) = 8, y’ (0) = 0
Find and graph or sketch the solution of the IVP. Show the details.y" + 4y’ + 5y = [1 - u(t - 10)]et - e10δ(t - 10), y(0) = 0, y’ (0) = 1
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.sin t(π/2 <
Find and sketch or graph f(t) if L(f) equals(1 + e-2π(s+1))(s + 1)/((s + 1)2 + 1)
Showing details, find f(t) if L(f) equals:
Using Theorem 3, find f (t) if L(F) equals:
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
What properties of matrix multiplication differ from those of the multiplication of numbers?
Are there any linear systems without solutions? With one solution? With more than one solution? Give simple examples.
Explain the use of matrices in linear transformations.
What is the role of rank in connection with solving linear systems?
Using the Laplace transform and showing the details of your work, solve the IVP:y''1 = -y1 + 4y2, y'2 = 3y1 - 2y2, y1(0) = 3, y2(0) = 4
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.cos2 ωt
What property of the Laplace transform is crucial in solving ODEs?
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - 1/4y = 0, y(0) = 12, y' (0) = 0
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.e2t sinh t
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.et (0 < t
Showing the details of your work, find L(f) if f(t) equals:t cos ωt
Find and graph or sketch the solution of the IVP. Show the details.y" + y = δ(t - π) - δ(t - 2π), y(0) = 0, y’ (0) = 1
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = y2 + 1 - u(t - 1), y'2 = -y1 + 1 - u(t - 1), y1(0) = 0, y2(0) = 0
When and how do you use the unit step unction and Dirac’s delta?
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.sin (ωt + θ)
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" + 7y' + 12y = 21e3t, y(0) = 3.5, y' (0) = -10
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.e-πt (2 < t
Showing the details of your work, find L(f) if f(t) equals:t2 cosh 2t
If you know f(t) = L-1{F(s)}, how would you find L-1{F(s)/s2}?
Find and graph or sketch the solution of the IVP. Show the details.4y" + 24y’ + 37y = 17e-t + δ(t - 1/2), y(0) = 1, y’ (0) = 1
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = 2y1 - 4y2 + u(t - 1)et, y2 = y1 - 3y2 + u(t - 1)et, y1(0) = 3, y2(0) = 0
Showing the details of your work, find L(f) if f(t) equals:te-kt sin t

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