Question

3(a) A beam of length ? is hinged at its ends x = 0 and x = ?. Suppose it carries a uniform load per unit length. Then for some particular load, the deflection z(x) satisfies the differential equation d^4z/dx^4 = 72, with boundary conditions z(0) = 0, z'(0) = 3?^3, z''(0) = 0, and z'''(0) = -36?. Solve this problem by direct integration of the differential equation and applying the boundary conditions at x = 0. (b) For the solution found in part (a), what are the values of z(?) and z''(?)? (c) Solve the problem of part (a) by using Laplace transforms. You may use the result that L{d^4z/dx^4} = s^4 Z(s) - s^3 z(0) - s^2 z'(0) - s z''(0) - z'''(0), where Z(s) = L{z(x)}. 4 Let f(x) be a function with period 2 and with f(x) = |x| for -1 < x < 1. Find the Fourier series for this function. You may use the result that ? x cos(n?x) dx = [cos(n?x) + n?x sin(n?x)] / (n^2 ?^2).

          3(a) A beam of length ? is hinged at its ends x = 0 and x = ?. Suppose it carries a uniform load per unit length. Then for some particular load, the deflection z(x) satisfies the differential equation

d^4z/dx^4 = 72,

with boundary conditions z(0) = 0, z'(0) = 3?^3, z''(0) = 0, and z'''(0) = -36?. Solve this problem by direct integration of the differential equation and applying the boundary conditions at x = 0.

(b) For the solution found in part (a), what are the values of z(?) and z''(?)?

(c) Solve the problem of part (a) by using Laplace transforms. You may use the result that

L{d^4z/dx^4} = s^4 Z(s) - s^3 z(0) - s^2 z'(0) - s z''(0) - z'''(0),

where Z(s) = L{z(x)}.

4 Let f(x) be a function with period 2 and with f(x) = |x| for -1 < x < 1. Find the Fourier series for this function. You may use the result that

? x cos(n?x) dx = [cos(n?x) + n?x sin(n?x)] / (n^2 ?^2).

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3(a) A beam of length ? is hinged at its ends x = 0 and x = ?. Suppose it carries a uniform load per unit length. Then for some particular load, the deflection z(x) satisfies the differential equation

d^4z/dx^4 = 72,

with boundary conditions z(0) = 0, z'(0) = 3?^3, z”(0) = 0, and z”'(0) = -36?. Solve this problem by direct integration of the differential equation and applying the boundary conditions at x = 0.

(b) For the solution found in part (a), what are the values of z(?) and z”(?)?

(c) Solve the problem of part (a) by using Laplace transforms. You may use the result that

Ld^4z/dx^4 = s^4 Z(s) - s^3 z(0) - s^2 z'(0) - s z”(0) - z”'(0),

where Z(s) = Lz(x).

4 Let f(x) be a function with period 2 and with f(x) = |x| for -1 < x < 1. Find the Fourier series for this function. You may use the result that

? x cos(n?x) dx = [cos(n?x) + n?x sin(n?x)] / (n^2 ?^2).

Added by Mary P.

Calculus: Early Transcendentals

James Stewart 8th Edition

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