I am trying to gain a level of fundamental understanding of an RC circuit sine wave response through the mathematics and was wondering if someone could help me work it out.

Fundamentally a sine wave is represented by the equation y=-ky'' . When a sine wave is used as the input of an RC low pass filter, you get a sine wave out (just phase and amplitude shifted). So I know that when the input waveform has the form Vin=-kVin'' then somehow the output also retains that relationship, Vout=-kVout''. I am having trouble proving this with the equation that describes an RC circuit response: Vout' = 1/RC*(Vin-Vout).

Any help would be great, thanks!