RC-Circuit With Two Capacitors that are neither in parallel nor in series
Perhaps this may help:



$$\begin{align*}
\frac{V_1}{R_1}+\frac{V_1}{R_2}+C_1\cdot\frac{\text{d}\,V_1}{\text{d}\,t}&amp;=\frac{V_s}{R_1}+\frac{V_2}{R_2}\tag{1a}\\\\
\frac{V_2}{R_2}+C_2\cdot\frac{\text{d}\,V_2}{\text{d}\,t}&amp;=\frac{V_1}{R_2}\tag{2a}
\end{align*}$$



$$\begin{align*}
&amp;\therefore\quad\text{the linear system of two 1st order diff-eqs is:}\\\\
\frac{\text{d}\,V_1}{\text{d}\,t}&amp;=\frac{1}{C_1}\cdot\left(\frac{V_s-V_1}{R_1}+\frac{V_2-V_1}{R_2}\right)\\\\
&amp;=\left[-\frac{R_1+R_2}{C_1}\right]\cdot V_1+\left[\frac{R_2}{C_1}\right]\cdot V_2+\left[V_s\cdot\frac{R_2}{C_1}\right]\tag{1b}\\\\
\frac{\text{d}\,V_2}{\text{d}\,t}&amp;=\frac{1}{C_2}\cdot\frac{V_1-V_2}{R_2}\\\\
&amp;=\left[\frac{1}{R_2\cdot C_2}\right]\cdot V_1 +\left[\frac{-1}{R_2\cdot C_2}\right]\cdot V_2+\left[0\right]\tag{2b}\end{align*}$$



$$\begin{align*}
&amp;\therefore\quad\text{where }\:a_{11}=-\frac{R_1+R_2}{C_1}, a_{12}=\frac{R_2}{C_1}, b_1=V_s\cdot\frac{R_2}{C_1}\\\\
&amp;\quad\quad\quad\quad\quad\: a_{21}=-\frac{1}{R_2\cdot C_2}, a_{22}=\frac{-1}{R_2\cdot C_2}, b_2=0, \\&amp;\quad\quad\text{then,}\\\\
\frac{\text{d}\,V_1}{\text{d}\,t}&amp;=a_{11}\cdot V_1+a_{12}\cdot V_2+b_1\tag{1c}\\\\
\frac{\text{d}\,V_2}{\text{d}\,t}&amp;=a_{21}\cdot V_1+a_{22}\cdot V_2+b_2\tag{2c}
\end{align*}$$



You can use whatever method you like. But the substitution method and the direct "guess and verify" methods come to mind. Either way, you want the homogeneous and the steady-state parts.

For one possible example, continuing with the homogenous portion:

$$\begin{align*}
\frac{\text{d}^2\,V_1}{\text{d}\,t^2}&amp;=a_{11}\cdot \frac{\text{d}\,V_1}{\text{d}\,t}+a_{12}\cdot\frac{\text{d}\,V_2}{\text{d}\,t}\\\\&amp;=a_{11}\cdot \frac{\text{d}\,V_1}{\text{d}\,t}+a_{12}\cdot\left(a_{21}\cdot V_1+a_{22}\cdot V_2\right)\\\\\text{from (1c) above (ignoring }b_2\text{)}, V_2&amp;=\frac{\frac{\text{d}\,V_1}{\text{d}\,t}-a_{11}\cdot V_1}{a_{12}}, \text{then,}\\\\
\frac{\text{d}^2\,V_1}{\text{d}\,t^2}&amp;=a_{11}\cdot \frac{\text{d}\,V_1}{\text{d}\,t}+a_{12}\cdot\left(a_{21}\cdot V_1+a_{22}\cdot \frac{\frac{\text{d}\,V_1}{\text{d}\,t}-a_{11}\cdot V_1}{a_{12}}\right)\\\\&amp;=\left(a_{11}+a_{22}\right)\cdot\frac{\text{d}\,V_1}{\text{d}\,t}+\left(a_{11}\cdot a_{22}-a_{12}\cdot a_{21}\right)\cdot V_1
\end{align*}$$



The above is a 2nd order diff-eq, in \$V_1\$, that can be now solved with the sum of two exponential expressions. With that in hand, the rest isn't hard.