Let u be the solution to the initial boundary value problem for the Heat Equation,du(t, x) = 5 d²u(t, x), t≤ (0, ∞), x = (0,3);0 and u(t, 3) = 0, and with initial conditionwith Dirichlet boundary conditions u(t,0)=u(0, x) = f(x)u(t, x) ==with the normalization conditions vn (0) = 1 and wn(2³2)∞x,The solution u of the problem above, with the conventions given in class, has the formn=10,= [0, ³²),3x =x Een vn (t) wn(x),= 1. Find the functions vn, wn, and the constants Cn.

Answered: Image /qna-images/answer/176d4ff3-0bbe-4732-8c07-8317a700be04.jpg

Answered: Let u be the solution to the initial…

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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Consider applying the method of separation of variables with u (x,t) = X (x) T (t) to the partial differential equation: (∂2u/∂x2) + (∂u/∂x) = (∂u/∂t) Which option gives the resulting pair of ordinary differential equations (where µ is a non-zero separation constant). A. X" (x) + X' (x) = µ, T. (t) = µB. X' (x) + X" (x) = µX (x), T.. (t) = µT (t) C. X' (x) = µ, T.. (t) + T. (t) = µD. X' (x) = µX (x), T.. (t) + T. (t) = µT (t) E. X" (x) + X' (x) = µX (x), T. (t) = µT (t)
11. (a) Show that is a solution of the system satisfying the initial condition X = sin t - t cos st) t sin t x² - (- ₁₂ ) x + (2 m) = x(0) = (1).
Find the most general functions X(x) and Y(y), each of one variable, such that Z(x, y) = XY satisfies the partial differential equation a²z Əz = Əx² дх2 - ду Obtain a solution of the above equation which satisfies the boun- dary conditions: when x = 0 or R Z=0 z = sin 3x when y = 0 and 0 < x < π.
The interval in which the solution of the given IVP is certain to exist (4-t2)y'+2ty 3t2, y(1)=5 Select one: a. (-2,2) b. (0,1) C. (0,2) d. NOTA e. (1,2)
Verify that any function of the form Y (t) = c1vt + czt* satisfies the equation ty"(t) – ĮtY'(t) + 2Y (t) = 0. Determine the solution of the equation with the boundary conditions Y(1) = 1, Y(4) = 2.
Let y = 0, +aqt+azt? +az13+aq +agt5+ag!6+ +а.t+ be the solution of d?y. +taY +y=tan-t ; y(0)=1, y'(0)= - 1 dt dt2 1 t=t- 3 - 1 tan 1 It +...** Write round your answer to three digits after the decimal sign.
Suppose f(x, y) g(x)h(y), where g and h are continuously differentiable functions of one variable with g(1) = 3, g'(1) =2, h(2) = 5, and h'(2) =-1. Use linearization to approximate f(1.1, 2.2).
Given yı (t) = t and y2(t) = t satisfy the corresponding homogeneous equation of t'y" – 2y = 1 – 2t°, t > 0 Then the general solution to the nonhomogeneous equation can be written as y(t) = c1y1(t) + ©2Y2(t) + Yp(t). Use variation of parameters to find a particular solution yp(t). Y,(t) = Tip: Before you use the formula Y29(t) Yı9(t) JW(n, y2) Yp = - Y1 + y2 W (y1, Y2) the ODE should be in the form y'' + a(t)y' + b(t)y = g(t)
Find a general solution to each linear ODE, and then find the specific solution with the given initial condition by using the integrating factor (a) u'(t) = u(t)+3, u(0) = 3 (b) u'(t) = 2u(t)+4, u(0) = 0 (c) u'(t) = −3u(t) +3, u(0) = 5 (d) u'(t) = −3u(t) +9t, u(0) = 5 (e) u' (t) = u(t)+2sin(t), u(0) = 1 (f) u'(t) = −4u(t) + e¹, u(0) = 2 (g) u'(t)= tu(t)+t, u(0) = 2 (h) u' (t) = u(t)/t+2, u(1) = 3 (i) u' (t) = sin(t)u(t) + sin(t), u(0) = 4 (j) u' (t) = au(t) +b, u(0) = uo, where a, b, and uo are constants

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