1- Solve the problemuz = Uxx + sin(nx) + sin(2nx)fu(0, t) = 0lu(1,t) = 0u(x, 0) = 0P.D.E.0

A partial differential equation is given by, ut = uxx+sinπx+sin2πx with boundary conditions u0, t =…

Answered: 1- Solve the problem P.D.E. Uz = Uxx +…

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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Let x (t) x' (t) x' (t) If x (0) x₁(t) = = = = = sin( x ₂ (t) 2 = sin( x1(t) x2(t) ] Put the eigenvalues in ascending order when you enter x₁(t), x₂(t) below. -17x₁(t) 8 x ₁ (t) 1 -3 [:] sin( t) + sin( t) + t) + cos( be a solution to the system of differential equations: and x'(0) t) + + cos( 16 x ₂ (t) 2 7x₂(t) = -28 16 cos ( t) + t) cos( t) + t) " find x(t).
Solve the equation Upx + Uyy = 0, 0
Find the only real eigenvalue of the problem x²y" - Xxy' + Xy y(1) = 0, y(2) — y' (2) - = 0, 1 < x <2 = 0, and find the corresponding eigenfunction.
6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
2. Use the Laplace transform to solve the initial value problem (a) x' - 5x = e-2t, x(0) = 3 ne lettre rouba23 (b) x' + 3x = 3t+7, x(0) = 1 (c) x" + 3x + 2x = 12et, (d) r" - 4x = t, on x(0) = 2, x(0) = 1, x'(0) = 3 x'(0) = 2.
2n+1 If x,=cos TEX, are the eigenfunctions for the following IBVP +co 0
Correct and detailed answer will be Upvoted else downvoted. Thank you!
Solve the initial value problem using Laplace Transform Method y" - 4y = sin 2t y(0) = 3, y'(0) = 0 40 e4t -8+ 212 + 10cos 2t +- y(t) sin 2t 11 40 e-4t + 10sin 2t + 212 5 y(t) = -8+ Cos 2t 11 121 -e4t + 40 1 yAt) = 1 Cos 2t + 10 11 sin 2t 5 8 1 121 e 40 1 11 yt): 8 - sin 2t + - cos 2t 10 5
I 5. Find all eigenvalues and eigenfunctions to the boundary value problem. Show all necessary work: y" + Ay = 0, y'(0) = 0, y' (L) = 0
part 2) a) Solve y" -6y' +9y=0. b) Find the particular solution of y"-6y' +9y=3-t c) Use the Laplace transform to solve the given initial value problem y"-6y' +9y=3-t, y(0)=0, y' (0) = 2
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1- Solve the problem P.D.E. Uz = Uxx + sin(nx) + sin(2nx) 0