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My professor started our PDE I class in Chapter six, so I am having a hard time with the really basic stuff to get the theory down.

One of my questions to answer is to verify a solution by using direct substitution.

[tex]u(x,t) \ = \ \frac{1}{2}\left[\phi(x+t) \ + \ \phi(x-t) \right] \ + \ \frac{1}{2} \int^{x+t}_{x-t}\Psi(s)ds [/tex]

With initial conditions

[tex] u(x,t_{0}) = \phi(x) \ , \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x), \ and \ t_{0} = 0 [/tex]

satisfies [tex]\frac{\partial^{2}u}{\partial x^{2}} - \frac{\partial^{2}u}{\partial t^{2}} = 0 [/tex]

It was easy for me to plug and chug to show that [tex] u(x,t_{0}) = \phi(x) \ and \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x) [/tex]

Clearly my next step is to find [tex] \frac{\partial^{2}u}{\partial x^{2}} [/tex]

but that's the step on which I'm stuck. Can someone get me started? If someone can show me how to do the second partial w.r.t x, it would be a good exercise for me to figure out the second partial w.r.t t. I apply the chain rule and just get a bunch of garbage back, which means I'm messing it up somewhere.

Any input is appreciated.

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# Wave Equation, stuck on a partial calculation

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