Solving Boundary Value Problem of a Wave Equation with Variable Coefficients

In this work, we solve the problem of boundary value of a wave equation with variable coefficients Where is the functions that represents the wave, and are continuous functions on the interval , and for any . That satisfies the boundary conditions , and initial conditions , solving the problem using the method of separating variables was difficult. Therefore, we found another way, which is to replace the wave equation with a system of first- order partial differential equations then solve it and we show that the solution of this system equivalent to the solution of a wave equation, using this method very helpful in getting a solution for this problem. Keywords: Wave equation, boundary conditions, initial conditions, system of first-order partial differential equations.