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This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with expectation and variance of a nonstationary process whereas our accompanying work deals with covariance and spectral density of a stationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the expectation and variance in the form of two ordinary differential equation (ODE) systems. They are derived within DSP theory without any techniques directly related to stochastic differential equations. Both ODE systems allow for space nonhomogeneities of the diffusion and damping matrixes and thereby do take nonlinearities of the DSP coefficients into account. Some related topics like invariant processes and the aspects of practical implementation of the above expressions are discussed as well. A proper attention is paid to formulation of some features important in applications of DSPs to the real-world problems. The results of this work can equally be used in various engineering fields.