Let u be the solution to the initial boundary value problem for the Heat Equation,du(t, x) = 5 du(t, x), t≤ (0, ∞), x = (0,3);with Neumann boundary conditions du(t,0) = 0 and du(t, 3) = 0 and with initial conditionu(0, x) = f(x) =Co =Cn =u(t, x)The solution u of the problem above, with the conventions given in class, has the form+29n==with the normalization conditions v₁ (0) = 1 and wn (0)vn (t) =wn(x)Co23,4,x = 0,x ECn Un (t) wn(x),= 1. Find the functions Un, wn, and the constants co and cn for n > 1.ΣMMΣ

In the problem heat equation is given. We solve the problem by method of separation of variable. In…

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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Given yi (t) t² and y₂(t) = t¹ satisfy the corresponding homogeneous equation of t'y" - 2y = -2t¹-3, t> 0, the general solution to the nonhomogeneous equation can be written as y(t) = y(t) + C₁y₁(t) + C2y2(t). Use variation of parameters to find y, (t). Yp(t) Preview
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.
2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x²e¯x. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mi = (iii) Assume the particle starts in xo energy Eo of the particle. Using (ii), with = F) O with positive initial velocity vo > 0. Find the initial show that the particle reaches x = 2 only if vo > û, 8e-2 = " m and in this case the particle's velocity in x = 2 is 8e-2 v(2): = v m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > û. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…
Let y=v,(x) y,+ v,(x) y; be the solution of y"+p(x) y+q(x)y3 In(x), which obtained by the variation of parameters method. If y, ()=x, and y,(x)=17 find v,(x). 33 vzK) =7 x² ( - - Intx)) 17 Vz(x) = x +In(x*) 17 None 35 v,(x) = x² (Inx -) Gear my choice
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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)
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