Thanks a lot! It sounds silly but I haven't heard of the standard method yet. OK I will try to use substitutions followed by using complex number (Euler's equation is useful!) to get a neat and mistake-free derivation.
I have learned in 1st year that the under-damped simple harmonic motion can be described by the differential equation m \frac {d^2 x} {dt^2} + b \frac {dx} {dt} + kx = 0 where m is the mass, b is the constant of linear drag and k is the spring constant
But the derivation is skipped...