 Much more general solutions are obtained by a generalization of an exact [[ordinary differential equation#solutions|particular solution]] derived by [[Edward Kasner]] for a field in vacuum, in which the space is homogeneous and has a [[Euclidean metric]] that depends on time according to the [[Kasner metric]]

("dl" is the [[line element]]; "dx", "dy", "dz" are [[infinitesimal]] [[Differential (infinitesimal)|displacements]] in the 3 [[spatial dimensions]], and "t" is time period passed since some initial moment "t"0 = 0).Here, "p"1, "p"2, "p"3 are any 3 numbers that satisfy the following "Kasner conditions"

Because of these relations, only 1 of the 3 numbers is [[independent variable|independent]] (2 [[equations]] with 3 [[unknown (mathematics)|unknowns]]).All 3 numbers are never the same; 2 numbers are the same only in the [[set (mathematics)|set]]s of values formula_30 and (0, 0, 1).