Ok, so I am dealing with a critically damped oscillator in which the natural frequency(w) of the oscillator is equal to the coefficient of friction (y). I am given the force mfe^t and told to find a solution for x, where x'' +2yx' +w^2 =fe^t. How do I go about doing this? The solution that I am supposed to find is Afe^t where A=f/4 I have to solve this for f=mfe^-t also, if this requires a different strategy, let me know I guess.
It would help a lot if you would clarify what you are saying. There is clearly a typo in your equation: it should be x'' +2yx' +w^2x =fe^t. But the main problem is that you seem to be using "f" to mean at least two different things. You say "I am supposed to find is Afe^t where A=f/4". Is that f<sup>2</sup>e<sup>t</sup>? But then "I have to solve this for f=mfe^-t". Surely f doesn't mean the same thing on both sides of that equation (since me<sup>-t</sup> is not 0!).