验证下列函数满足拉普拉斯方程uxx+uxy=0: (1)u=arctanx/y; (2)u=sinx×coshy+cosx×sinhy; (3)u=e-xcosy-e-ycosx.

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问题 验证下列函数满足拉普拉斯方程uxx+uxy=0:
(1)u=arctanx/y;    (2)u=sinx×coshy+cosx×sinhy;    (3)u=e-xcosy-e-ycosx.

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答案[*] 综上可得uxx+uyy=0,此函数满足拉普拉斯方程. (2)ux=cosx×coshy-sinx×sinhy uxx=-sinx×coshy-cosx×sinhy uy=sinx×sinhy+cosx×coshy uyy=sinx×coshy+cosx×sinhy 综上可得uxx+uyy=0,此函数满足拉普拉斯方程. (3)ux=-e-xcosy+e-ysinx uxx=e-xcosy+e-xcosx uy=-e-xsiny+e-yycosx uyy=-e-xcosy-e-ycosx 综上可得uxx+uyy=0,此函数满足拉普拉斯方程.

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