RLC circuit equation

Question

I want to make equation that describe relation between vi and vc. I did it like this. Is this right way?

Answer 1

To help a little (check results of differential equations), here are pictures of the inductor current and the voltage across the capacitor ...
Values : L:= 1e-3 H: C:= 1e-6 F: r1:= 1 ohm : r2:= 100 Ohm: Vin:=step 1 V.

Answer 2

Well, using Laplace transform we can see that:

$$\mathscr{H}\left(\text{s}\right):=\frac{\text{V}_\text{out}\left(\text{s}\right)}{\text{V}_\text{in}\left(\text{s}\right)}=\frac{\text{R}_2\space\text{||}\space\frac{1}{\text{s}}}{\text{R}_1+\text{sL}+\left(\text{R}_2\space\text{||}\space\frac{1}{\text{s}}\right)}\tag1$$

Where \$\alpha\space\text{||}\space\beta:=\frac{\alpha\beta}{\alpha+\beta}\$.

So, when we simplify we can write:

$$\text{R}_2\space\text{||}\space\frac{1}{\text{s}}=\frac{\text{R}_2\cdot\frac{1}{\text{s}}}{\text{R}_2+\frac{1}{\text{s}}}=\frac{\text{s}}{\text{s}}\cdot\frac{\text{R}_2\cdot\frac{1}{\text{s}}}{\text{R}_2+\frac{1}{\text{s}}}=\frac{\text{R}_2\cdot\frac{\text{s}}{\text{s}}}{\text{sR}_2+\frac{\text{s}}{\text{s}}}=\frac{\text{R}_2}{1+\text{sR}_2}\tag2$$

So, we get:

$$\mathscr{H}\left(\text{s}\right)=\frac{\text{R}_2}{1+\text{sR}_2}\cdot\frac{1}{\text{R}_1+\text{sL}+\frac{\text{R}_2}{1+\text{sR}_2}}=\frac{\text{R}_2}{\text{R}_2+\left(\text{R}_1+\text{sL}\right)\left(1+\text{sR}_2\right)}\tag3$$