So this circuit is supposed to model a bipolar junction transistor. I am a bit stuck with solving for the ratio listed because V0 tends to go away in my attempts. 1. I start by writing the current through the diode depending entirely on the voltage supplied by Vs. 2. With IE in terms of Vs I can multiply with alpha and RL to get the voltage across the resistor. 3.Then I write Vs and VL in terms of their DC and AC components. Then I get that because V0 << VT I can treat that exp(VoCos(wt)/VT)=1 and it drops away. What I would expect for this circuit is for the ratio between V0 and V2 to be equal to the ratio between vB and v1. The intent of the circuit I imagine would be to provide a gain to a small signal (here that is the AC term) while the DC term maintains the diode's forward bias. Is there a better way to approach this problem?
You are making too much of an approximation. You are assuming the exponential term is a constant. What you need to do is keep the constant and the linear term.
By telling you that V0 << Vt (thermal voltage ~= 26mVdc @ room temp.), this ends up being a simple small-signal theory problem. The input signal contains what looks like a DC term Vb and a small-signal V0cos(wt). You can approximate a linear relationship between the current/voltage through/across the diode by assuming that V0 is a very small AC signal wiggling around a DC signal Vb. Given the equation of the diode, you can differentiate to find the slope, to determine how much the current will fluctuate as a result of the voltage perturbation. You can drop the '-1' term for the differentiation to simplify the result as it is insignificant.