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If there wasn't that minus sign, the answer would be a wave equation. http://uniquation.com/ was a bust. I asked wolframalpha, and it came back with an answer which looked just like the wave equation with an extra factor of $i$. Does the equation have a more general name?

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It is a Laplace-type equation. –  André Nicolas Oct 6 '11 at 2:14
    
The question came up in a 4D context, specifically: u_tt + u_xx + u_yy + u_zz = 0, so that would be a 4D Laplace equation. The inhomogeneous case would be 4D Poisson. Thanks. –  sweetser Oct 6 '11 at 3:05
    
I'd like to add that the conic representation of this is an ellipse and the function is elliptic. –  Tyler Hilton Oct 6 '11 at 3:44
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2 Answers

The equation is of the Laplace type.

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If $u_{tt}=-u_{xx}$ are given the conditions of the type $u(x,t_1)$ , $u_t(x,t_1)$ , $u(x_1,t)$ and $u(x_2,t)$ , you will feel that it is unlike to solving the "laplace equation" and it is like to solving the "wave equation".

If $u_{tt}=u_{xx}$ are given the conditions of the type $u(x,t_1)$ , $u(x,t_2)$ , $u(x_1,t)$ and $u(x_2,t)$ , you will feel that it is unlike to solving the "wave equation" and it is like to solving the "laplace equation".

So whether $u_{tt}=-u_{xx}$ is belongs to the "wave-type equation" or "laplace-type equation" should be controversial, especially when the $t$ in here is not represent as time or $x$ in here is not represent as position, or either or both of them haven't any physical meaning.

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