How to find differential equation satisfied by \$u_{R_{equivalent}}\$, \$u_{L_{equivalent}}\$ or \$u_{C_{equivalent}}\$ in complicated RLC circuits?
In linear circuits (basically R,L,C and controlled sources), you can analyze at each individual frequency. Each frequency is a pure sinusoid, and because the response of each element to a sinusoid is also a sinusoid (at the same frequency), but with a different amplitude and/or phase, it is possible to use complex numbers to represent the behavior. Thus a capacitor's impedance becomes 1/(j.2.pi.f.C) where j = sqrt(-1). When an input signal is repetitive, it is possible to break this down to a collection of sinusoids, analyze and then combine the results from each frequency. 

This analysis can be performed symbolically, or numerically.

In non-linear circuits, this method does not apply, and in general it is not possible to solve the differential equations directly. Numerical approximation methods are used to solve them in simulators such as SPICE.