verify that general DAlembert solution satisfies the initial

verify that general D\'Alembert solution satisfies the initial condition u(x,0) and u_t(x,0)=g(x)

Solution

General D\'Alembert solution

u(x, t) = [F(x at) + F(x + at)] / 2,
where F(x) is odd periodic function of period 2T such that F(x) = f(x) on
the interval 0 < x < T,

so it will satisfy the given initial-boundary value
problem by checking.....

1)the wave equation,
2.)boundary conditions,
3.) initial conditions

a² u_xx = u_tt , 0 < x < T , t > 0,

1.) u(0,t) = 0, u(T,t) = 0,

2.)u(x, 0) = f(x) = 0,

3.)ut(x, 0) = 0.

and we know thtat f(x)=0 at time t=0

That\'s why u(x,0 )and ut(x,0) = 0   

VERIFIED