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It is known that the linear PDE $u_{xy}=u$ has this quite nice integral kernel form of general solution e.g. according to Method of charactersitics and second order PDE.:

$u(x,y)=\int_0^xf(s)I_0\left(2\sqrt{y(x-s)}\right)~ds+\int_0^yg(s)I_0\left(2\sqrt{x(y-s)}\right)~ds$

How about the linear PDE $u_{x_1x_2x_3......x_n}=u$ ?

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