3. Consider the heat equationduPukfor 0 < x < L, and t > 0with the boundary conditionsdu-(0, t) = 0 andди(L,t) = 0, for t> 0.Solve for the following initial value conditions:a) f(x) = 2+ 3 cos3TT)L(10€ (0, L/2]b) f(x) =21 (L/2, L)

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Answered: 3. Consider the heat equation ди k for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Consider the heat equation ди k- for 0 0 with the boundary conditions u(0, t) = 0 and u(L, t) = 0, for t > 0. Solve for the following initial value conditions: a) f(x) = 4 sin ( 1 b) f(x) = x € (0, L/2] x € (L/2, L)
a) A vibrating spring can be modeled by the initial value problem: mx" (t) + bx'(t) + kx(t) = 0 With x(0) — х, аnd x'(0) %3 Vo Solve the above equation when: 10kg, b = 12 kg/sec,k = 37 kg/sec? ,x, 3D 50ст апd v, 3D 10ст/sec m =
10. (6 pts) Find the most general antiderivative of f(r) = sinh r + sin r. 11. (8 pts) (a) If g(x) = (tant) dt, find g'(r). (b) If h(x) = (tan t)° dt, find h'(x).
Find the general solution to dw ol Sa'(t) = x(t) – y(t) + 3e'; ly'(t) = y(t) – x(t) + 2e'; (8) (8) %3D
Determine the longest interval in which the initial value problem is certain to have a unique twice differentiable solution. (Do not attempt to find the solution.) (t− 3)y" - 3ty' + 5y = sint, y( − 7) = 7, y'(-7) = 3 ◄►
The temperature u(x, t) of a metal bar of length e = 2 at a distance r from one end and at time t is modelled by the partial differential equation Ut = aurr (0 0) It is given that the metal has diffusivity a = 2.25, that the two ends of the bar are kept at temperature u = 0 and that the initial temperature distribution is u(x,0) = f(x) = sin(Tx/0) Use the explicit difference parabolic method with Ar = 0.5 and At 0.05 to approximate u(x, t) at t= 0.05 and t= 0.10.
4. (a) Solve the IVP for y(t): y" + 4y = 3 8(t – n), y(0) = 0, y' (0) = 0 (b) 2n. Be sure to label your graph so that the period and amplitude of any oscillations are clearly identified. Graph the solution y(t) from part (a) versus t, for values of t between 0 and
As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t) = 9-t N/m. If the unforced mass-spring system has mass m = 2 kg and a damping constant b = 1 N-sec/m with initial conditions x(0) = 2 m and x'(0) = 0 m/sec, then the displacement x(t) is governed by the initial value problem 2x" (t) +x' (t) + (9 − t)x(t) = 0; x(0) = 2, x'(0) = 0. Find the first four nonzero terms in a power series expansion about t = 0 for the displacement. x(t) = ☐ +... (Type an expression that includes all terms up to order 4.) k(t)=9-t mrr 1 N-sec/m 2 kg heat x(t) x(0)=2 x'(0)=0
Given equation is using La place 1- y" - 3y' + 2y = 4e²t y(0)= -3 & y'(0) = -5 2- y" - 3y' + 2y = e³t y(0)= 0 & y'(0)= 0 3- y" - 5y + 6 = sin t. y(0) = 1/10 & y'(0) = 21/10. 4- y" - 3y' + 2y = 4e²t y(0)= -3 & y'(0)= 5 5- y" - 4y = 2e³t y(0)= 1 & y'(0)= -1

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3. Consider the heat equation ди k for 0 < x < L, and t > 0 with the boundary conditions ди (0, t) = 0 and ди (L,t) = 0, for t > 0. Solve for the following initial value conditions: a) f(x) = 2+ 3 cos (3T) L (1 0€ (0, L/2] b) f(x) = 1 (L/2, L)