partial-differential-equations-by-evans · Problem 15

Nonlinear First-Order PDE

Mathematics — Calculus · wave-equation-dalembert

Problem

(a) Show the general solution of the PDE uxy=0u_{xy} = 0 is u(x,y)=F(x)+G(y)u(x,y) = F(x) + G(y) for arbitrary functions F,GF, G. (b) Using the change of variables ξ=x+t\xi = x+t, η=x−t\eta = x-t, show utt−uxx=0u_{tt} - u_{xx} = 0 if and only if uξη=0u_{\xi\eta} = 0. (c) Use (a) and (b) to rederive d'Alembert's formula.

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