Differential equation of a RRC circuit as Voltage output/voltage input?
You're right to be lost with this one because it looks like the presentation is giving an example in the time domain and then ending with a generic format in the frequency domain (also weird that the schematic takes the time to define 3 separate current paths but instead just uses "i").

Using an s-domain table you can see that the final transfer function is a convolution between a step input and an exponential which makes sense because any voltage applied at the circuit input will immediately cause the output to rise to the R1/(R1+R2) voltage point and then charge the capacitor to the full voltage source level (exponential).

If you want to find the transfer function yourself (Vout/Vin aka G(s)), I'd solve the voltage divider equation in the frequency domain where "C" becomes "1/(sC)" (voltage drop across "R" is not frequency dependent so it stays as "R"). If you calculate the output voltage using the frequency domain equivalents, you should be able to distill the equation to the same format as their transfer function "template" and get values for "K" and "T."

The analysis would look similar to this but with the added complexity of R1 in series with 1/(sC) in the numerator and denominator:
http://www.analogzoo.com/2015/12/deriving-the-rc-filter-transfer-function/