Question 4 Consider the following wave equation with a constant forcing term Pu du - 4, 0 0 and the boundary conditions and initial conditions are du (0, x) = sin(2T x) dt u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2x(x – 1), %3D %3| (a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should satisfy so that v(t, x) satisfies the standard wave equation (b) Solve the differential equation that you found in part (a) and find a solution (x) so that the boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions v(t, 0) = 0, v(t, 1) = 0. (c) For the solution (x) that you found in part (b), convert the initial conditions into new dv conditions for v(0, x) and at (0, 2). (d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nTx), find n=1 an, bn and hence, the final solution u(t, x).

Question 4 Consider the following wave equation with a constant forcing term Pu du - 4, 0 0 and the boundary conditions and initial conditions are du (0, x) = sin(2T x) dt u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2x(x – 1), %3D %3| (a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should satisfy so that v(t, x) satisfies the standard wave equation (b) Solve the differential equation that you found in part (a) and find a solution (x) so that the boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions v(t, 0) = 0, v(t, 1) = 0. (c) For the solution (x) that you found in part (b), convert the initial conditions into new dv conditions for v(0, x) and at (0, 2). (d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nTx), find n=1 an, bn and hence, the final solution u(t, x).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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