When i'm given simple RLC circuits i know how to find the expression for each voltage, for example the expression of \$u_R\$ the voltage between the borns of the resistor \$R\$, or the expression of \$u_C\$ the voltage between the borns of the capacitor \$C\$, or the expression of \$u_L\$ between the borns of the inductor \$L\$... it all starts by finding first the differential equation they satisfy usually using Kirchkov's law of currents. not only but also in complicated circuits like this one
http://sketchtoy.com/66481514 (someone please post image i can't do it (on ipad))
the basic idea is to find \$R_{equivalent}\$ and \$L_{equivalent}\$ and \$C_{equivalent}\$ and then to use same past technique, Kirchkov's law => differential equation => solution (expression desired). (what we would be searching then is an expression for the voltage between the borns of the equivalent resistor, and not each resistor)
but how to do when you have the resistors mixed up with capacitors in parallel and stuff like that ? (as below) how can we find the diff. equation satisfied by \$u_{R_{eq}}\$, \$u_{C_{eq}}\$, and \$u_{L_{eq}}\$, in that case?
Edit: I'm not looking in particular for the solution of these differential equations, just to find the differential equation itself.
(sorry for writing and links to images, using ipad to write is difficult)