A stretched string is a known source of a variety of linear and non-linear mathematical models. Vibrations of thin stretched string carrying an alternating electric current in a non-uniform magnetic field are described by nonlinear equations that have been considered partially in a few previous works, but not presented in a general form. In this work, for the first time, we present the complete mathematical model of oscillations of a current-carrying string in a magnetic field as a set of non-linear coupled integral-differential equations. This model includes three dominating physical effects governing string oscillations, adding to these a viscous damping. Parametric modulationParametric modulation of the natural frequency of the string caused by the thermic effect of an alternating current is responsible for the unlimited growth of the amplitude of oscillation. The increase of tensionTension due to the elongation of string while oscillating causes the opposite effect suppressing the unlimited growth of amplitude of string oscillation, this is represented in equations of the model as the cubic non-linearity. The interaction of the electric current in a string with the magnetic field gives rise to the alternating distributed nonuniform Lorentz force that causes resonancesResonances and instabilities in the string and non-linear coupling between the modes of different polarizations. In the frame of simplified model resulted in the Mathieu-Duffing type driven oscillator, we illustrate numerically the complexity of string oscillations caused by the combination of these effects.