![Question 4<br>Consider the following wave equation with a constant forcing term<br>Pu du<br>- 4, 0<x < 1, t > 0<br>-<br>and the boundary conditions and initial conditions are<br>и(t, 0) 3D 0, и(t, 1) 3D 0, и(0, х) —D 2л(т — 1),<br>du<br>(0, x) = sin(2T x)<br>(a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should<br>satisfy so that v(t, x) satisfies the standard wave equation<br>(b) Solve the differential equation that you found in part (a) and find a solution (x) so that the<br>boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions<br>v(t, 0) = 0, v(t, 1) = 0.<br>(c) For the solution (x) that you found in part (b), convert the initial conditions into new<br>dv<br>conditions for v(0, x) and<br>(0, x).<br>(d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nrx), find<br>n=1<br>an, bn and hence, the final solution u(t, x).<br>](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_5gz6xz0-3kgs1y5e.jpeg)
Extracted text: Question 4 Consider the following wave equation with a constant forcing term Pu du - 4, 0
< 1,="" t=""> 0 - and the boundary conditions and initial conditions are и(t, 0) 3D 0, и(t, 1) 3D 0, и(0, х) —D 2л(т — 1), du (0, x) = sin(2T x) (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 (0, x). (d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nrx), find n=1 an, bn and hence, the final solution u(t, x).