Electrical – Difference between the natural response and transient response of an RLC circuit

The step response of your circuit will normally be composed a constant term and exponentials, and the exponentials can take various forms, depending on the relative values of the components.

For example, for relatively large values of resistance the exponentials will be of the form: \$ e^{-\sigma t}\$; \$ e^{-\beta t}\$, which decay to zero over time.

A sine wave, if it exists, arises from Euler's Formula: \$\small e^{jx}=cos(x)+jsin(x)\$.

Thus, for smaller resistance values the exponentials may have complex powers: \$ \small e^{-(\sigma \pm j\omega )t}\$ and these cases can be expressed as: $$ e^{-(\sigma \pm j\omega )t}=e^{-\sigma t} \: e^{\pm j\omega t} = e^{-\sigma t}[cos(\omega t) \pm j\:sin(\omega t)]$$ which is an exponentially decaying sinusoid.

It's interesting to note that if the resistance is zero, \$\small \sigma =0\$, and the response is a pure sine wave.

You're not meant to have 2 independent sources working at the same time in the circuit, so the best way to deal with this is to use source transformation which will produce a voltage source making the circuit a combination of 2 voltage sources. I'm not sure it can be solved otherwise. Besides before you solve for neper frequency the RLC circuit has to be either a completely parallel or series circuit. Your circuit above is neither (series-parallel), to find 'alpha' you have to use source transform to make it a completely series circuit.