Finding the voltage of a capacitor
I have a little different opinion from your prof. The characteristic equation of an RLC circuit will be either of the two given below.
$$s^2 + \frac{R}{L}s +\frac{1}{LC} = 0\tag1$$
$$s^2 + \frac{1}{RC}s +\frac{1}{LC} = 0\tag2$$
Reason:
The characteristic equation of a second order system is
$$s^2 + 2\alpha s + w_0^2 = 0$$
Where \$\alpha\$ is a damping related parameter and \$w_0\$ is the natural undamped frequency.
For an RLC circuit, \$w_0\$ will depend only on L and C and it will be \$\frac{1}{\sqrt{LC}}\$.
Coming to the dimension of the characteristic equation, each term in it should have a dimension of \$\mathrm{radian^2}\$. So \$2\alpha\$ must have dimension of \$\mathrm{radian}\$. The only possible options are \$\frac{1}{RC} \$ and \$\frac{R}{L}\$.
Which equation applies where?
Equation (1) applies to RLC circuits in which two more components are in series. The two possible connection are:

The R L and C can take any of these positions Z1, Z2 and Z3, but only one position at a time.
Equation (2) applies to RLC circuits in which two more components are in parallel. The two possible connection are:

The verification of this result is left to the reader. :-)
In the circuit given in the question, two components R and L are in series and hence the corresponding characteristic equation will be (1).