Answer to Question #263434 in Differential Equations for Rubi

Question

Determine the general solution to the equation ∂²u/∂t²=c²(∂²u/∂x²) under the boundary conditions u(0,t)=u(1,t)=0 and initial conditions u(x,0)=Φ(x), u_t(x,o)=Ψ(x)

Answer 1

solution of the one-dimensional wave equation:

$u(x,t)=\sum A_nsin(\pi nx/l)cos(\pi nct/l)$

where

$A_n=\frac{2}{l}\int^l_0 u(x,0)sin(\pi nx/l)dx=\frac{2}{l}\int^l_0 \Phi (x)sin(\pi nx/l)dx$

then:

$u_t(x,t)=-\frac{A_n\pi c}{l}sin(\pi x/l)sin(\pi ct/l)$

$u_t(x,0)=Ψ(x)=0$

l is length of object where wave occurs (for example, string)