Partial differential equation

D'Alembet's solution for one-dimensional wave equation ( homogeneous)

Consider the initial value formula 

(¶^2 u)/(¶x^2 )=1/c^2    (¶^2 u)/(¶t^2 )   -∞<x<∞    t>0

Satisfying the initial conditions u(x,0)=f(x),  ∂u/∂t (x,0)=g(x)

This problem is known as the Cauchy problem for

 one-dimensional wave equation. 

D’Alembert’s solution for one-dimensional wave

equation (non-homogeneous )

Consider one-dimensional wave equation  u_tt-c^2 u_xx=F(x,t)

-∞<x<∞    t>0 . u(x,0)=f(x), u_t (x,0)=g(x) -∞<x<∞

Where F(x,t) is the source acting on an incoming string