Math Help - how do i show something is a solution without solving it? 2nd order DE

1. how do i show something is a solution without solving it? 2nd order DE

the wave equation

δ2y/δx2 = (1/c2)(δ2y/δt2)

show y(x,t) = f(x-ct)+g(x+ct) is a solution explicitly

please show me how, don't just say "substitution"

2. Re: how do i show something is a solution without solving it? 2nd order DE

don't just say "substitution"
Ahem...

Anyway, just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x and t in turn), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Compare bottom rows.

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!