Transient response for a second-order system.

A student asked me how to find the time delay, t_d  for a second-order system by applying mathematical equation.

Let c_{final} denote the final value of the waveform. Then, the rise time t_r will be the time duration between 0.1c_{final} to 0.9c_{final} .

Since the time delay t_d equals to the time for the waveform to reach 0.5c_{final}. My rough idea for the mathematical equation is:
t_d \approx 1.25 * t_r / 2 .

Just an approximation as the waveform between 0.1c_{final} to 0.9c_{final} is not a linear line.