Let u(x,t)=ϕ(x+ct)+ψ(x−ct) where ϕ and ψ are arbitrary functions. Find ∂x∂u and ∂t∂u in terms of the partial derivatives of ϕ and ψ.I've written that
u=u(ϕ,ψ) and
ϕ=ϕ(x,t),ψ=ψ(x,t) and using the chain rule I ended up with...
∂x∂ut=
∂ϕ∂uψ∂x∂ϕt+∂ψ∂uϕ∂x∂ψt∂t∂ux=
∂ϕ∂uψ∂t∂ϕx+∂ψ∂uϕ∂t∂ψxIs that it? The question asks for an answer in terms of
partial derivatives of
ϕ and
ψ but I still have partial derivatives of
u... I don't know how to get rid of that though since I don't know what
ϕ and
ψ are. And my answer has nothing to do with the fact that
ϕ and
ψ are functions of
x+ct and
x−ct so I'm getting the feeling I'm missing something... but I'm not sure what. Could someone give me a hint? Thanks!