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ouchimdead's Question from Math Help Forum

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Title: 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"
Hi ouchimdead, :)

The only thing that you have to do here is to find the second derivatives of \(y(x,t)\) with respect to \(x\) and \(t\). Then it could be easily seen that they satisfy the given differential equation. :)

\[y(x,t) = f(x-ct)+g(x+ct)\]

\[\Rightarrow\frac{\partial}{\partial x}y(x,t) = \frac{\partial}{\partial x}f(x-ct)+\frac{\partial}{\partial x}g(x+ct)\mbox{ and }\frac{\partial}{\partial t}y(x,t) = -c\frac{\partial}{\partial t}f(x-ct)+c\frac{\partial}{\partial t}g(x+ct)\]

\[\Rightarrow\frac{\partial^{2}}{\partial x^2}y(x,t) = \frac{\partial^{2}}{\partial x^2}f(x-ct)+\frac{\partial^2}{\partial x^2}g(x+ct)\mbox{ and }\frac{\partial^2}{\partial t^2}y(x,t) = c^2\frac{\partial^2}{\partial t^2}f(x-ct)+c^2\frac{\partial^2}{\partial t^2}g(x+ct)\]

\[\therefore \frac{\partial^{2}}{\partial x^2}y(x,t)=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}y(x,t)\]

Kind Regards,
Sudharaka.