I have an RC circuit as shown below
unfortunately I have no knowledge about RC solving but I want to obtain corresponding differential equations for V1, V2 and V3. Having in hand differential equation for V1 is enough for now, if anyone can tell me it!!!
simulate this circuit – Schematic created using CircuitLab
In box 1, we're converting our current source into a voltage source and further converting it to its s-domain equivalent.
Then in box 2 we're taking the s-domain source together with the s-domain equivalent of the impedances of the given circuit. (see this)
Assuming that the capacitances had no voltage across them initially, (i.e. \$V_{0_{C_1}}=V_{0_{C_2}}=0\$, the s-domain circuit is equivalent to that in box 3 where \$V=-V_1\$ and \$R_{\text{equiv}}=\left[\left[\left(R_2+\frac{1}{sC_2}\right) \parallel\ R_L \right]\parallel \frac{1}{sC_1}\right]+R_1=R_1+\left[\frac{1}{R_2+sC_2}+\frac{1}{R_L}+\frac{1}{sC_1}\right]^{-1}\$
Hence $$V_1=-V=-\left(\frac{R_\text{equiv}}{R_s+R_\text{equiv}}\right)\cdot\frac{iR_s}{s}$$
The expression for \$V_1\$ will be a function of \$s\$; applying inverse laplace transform (using tables is easier) to the equation will give you the differential equation you require.
