This paper puts forward a new method of solving the problem for calculating the unknown and non-measurable parameters, that are included in a system of differential equations, whose solution adequately reproduces the given experimental data, but has no analytical form. The problems of this kind are often found in physical research of semiconductor materials, biomedical processes and in electronics. The novelty lies in the proposed idea of numerical calculations of partial derivatives which has made it possible to adapt the Levenberg – Marquardt method of non-linear approximations for solving the said problem. Our specific examples showed that calculation errors of the parameter values were not more than the experimental errors.