Consider the circuit
The writing in red are my own additions (so let me know if it's wrong).
From Kirchhoff's Laws, I obtained
\$I=I_{2}+I_{1}\$
\$I_{1}=I_{3}+I_{4}\$
\$V=V_{C_{1}}=V_{R_{2}}=V_{C_{2}}\$
\$V=R_{1}I=R_{3}I_{4}\$
\$I_{2}=C_{1}\frac{d}{dt}V_{C_{1}}\$
\$V_{R_{2}}=R_{2}I_{3}\$
\$I_{4}=C_{2}\frac{d}{dt}V_{C_{2}}\$
Did I get it right? Again, I'm not really sure as I'm not entirely familiar with circuits and the sort, being a mathematician rather than an electrical engineer.
In the end I want to come up with a differential equation that links \$V\$ with \$V_{C_{2}}\$, but I imagine that I can reach that via elimination, provided I have these equations right.
Best Answer
I think it should be like this:
\$I=I_{2}+I_{1}\$
\$I_{1}=I_{3}+I_{4}\$
\$\$
\$V=V_{C_{1}}+V_{R_{1}}\$
\$V=\frac{1}{C_{1}}\int I_{2}\mathrm{d}t+R_{1}I\$
\$\$
\$V_{R_{2}} = V_{C_{1}}\$
\$R_{2}I_{3} = \frac{1}{C_{1}}\int I_{2}\mathrm{d}t\$
\$\$
\$V_{R_{2}}=V_{R_{3}}+V_{C_{2}}\$
\$\$
\$V_{R_{3}}+V_{C_{2}}=V_{R_{2}}\$
\$R_{3}I_{4}+\frac{1}{C_{2}}\int I_{4}\mathrm{d}t=R_{2}I_{3}\$
The rest should be a math excercise, I think :-)