An alternative approach to the one-dimensional wave equation is to recast the pde as a pair of ode. consider the wave equation with forcing term,

∂2u/∂t2−c2∂2u/∂x2=f.

a vector-valued function v=(v1,v2) with components v1: =∂u/∂t, v2: =∂u/∂x.

v satisfies a vector equation (4.49) ∂v/∂t−a·∂v/∂x=b, where b: =(f,0) and a is the matrix a: = [0 c2 1 0].

the vector equation (4.49) can be solved by diagonalizing a.

if we set t: =[1 c 1 −c], then tat-1= [c 0 0 −c]. under that the substitution w: =tv, equation (4.49) reduces to a pair of linear conservation equations for the components of w: {∂w1/∂t−c∂w1/∂x=f, ∂w2/∂t+c∂w2/∂x=f. equation (4.50)

(c) translate the initial conditions u(0,x)=g(x), ∂u/∂t(0,x)=h(x) ,into initial conditions for w1 and w2, and then solve (4.50) using the method of characteristics. combine the solutions for w1and w2 to compute v1=∂u/∂t, and then integrate to solve for u. your answer should be a combination of the d’alembert formula and the duhamel formula.