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Spherical waves in odd-dimensional space. (English) Zbl 0684.35063

The author gives the general solution of the \((2n+1)\)-dimensional wave equation with spherical symmetry \[ u_{tt}-u_{xx}+(2n/x)u_x=0, \quad n \text{ is an integer}, \] in terms of two arbitrary functions and their first \(n\) derivatives. Transformations then yield the general solutions to the Euler-Poisson-Darboux equation \[ u_{xy}- (n/(x+y))(u_x+u_y)=0 \] and the one-dimensional wave equation \[ u_{tt}-x^{4n/(2n+1)}u_{xx}=0. \]

MSC:

35L05 Wave equation
35C05 Solutions to PDEs in closed form
35Q05 Euler-Poisson-Darboux equations
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