Partial differential equation Homework Help

A violin string produces a vibration where we have that A > 0, that a and b are arbitrary constants, and that utt = c²uxx for some c. Find E>0 in terms of A. E=

Decide whether following functions satisfy the wave equation utt =a² uxx.

4 If w = u² cos(v) where u(t) = 3 e^t and v(t) = 3 t³, find ·

a. Find ztt for z(x, y, t) = cos(√9+16t) sin(3x) sin(4y). b. Does u = sin(√25 t) sin(3x) sin(4y) satisfy the membrane equation utt = uxx + uyy ?

2. A support for electrified railway cables is cantilevered from the side of the track by a beam with spring stiffness k. The mass of the beam is 3M and is assumed to be concentrated at the free end. The cable, of mass 2M, is supported by a spring of stiffness k from the end of the cantilever. The system of equations governing the motion of the system is: 3 M y_{1}^{\prime \prime}=-2 k y_{1}+k y_{2} 2 M \ddot{y}_{2}=k y_{1}-k y_{2} k = 22 Write the above system of differential equations in matrix form. Then, by considering the trial solution: y = e"X, show that system can be written as an eigenvalue problem. (3) b) Find the general solution for the system of equations by solving the eigenvalue problem. (12)

An infinite solid circular cylinder is initially at a uniform temperature of 150° C. At time t = 0 the temperature around the entire boundary is suddenly reduced to 0°C, and maintained thereafter. Determine the temperature at any point of the region at any subsequent time.

1. General solution of the wave equation using Fourier Transforms. Define the Fourier Transform pair as (note the sign difference in the definition from that of Haberman, § 10.3.2) ƒ(k) = f(x)e ikzdr f(x) (a) Show that the general solution of (§ 10.6.1 of Haberman) Pu Ət² f(x), is 1 ikr 2/7 f(k) e³kx dk, 2π 2 Pu əx² u(x,0) (-∞0<x<∞) Ju(x, 0) Ət u(x,t) = ½ [ƒ(x − ct) + f(x + ct)] 0 (1) (2) (3)