Where is the Error in this AC/DC Circuit Analysis?

[pierce15]

See picture, sorry that it's huge. V1 is a DC voltage and there is also an oscillating source of frequency omega. Let V be the voltage between the two resistors, I be the current exiting the cap, and I1 and I2 be the currents through the 2 resistors. Then we have: $I_1 = (V - V_1) / R_1$ , $I_2 = V / R_2$, $I_1 + I = I_2$, and $I = - C \frac{d}{dt} (V - V_0 \cos (\omega t))$. Combining gives $-C \frac{d}{dt} (V - V_0 \cos (\omega t )) = V / R_2 - (V - V_1)/ R_1 = V( 1/R_2 - 1/R_1) + V_1 / R_1$. The constant term can be eliminated with a substitution, and then the sign on the right hand side can be chosen to give a homogenous solution of the form $e^{ax}$ with $a > 0$, by choosing $R_1$ accordingly. This obviously makes no sense. Can someone see where I am setting this up wrong?

 

[mfb]

The sign or direction of I₁ looks inconsistent.

 

[pierce15]

Sorry for not updating. The current across R1 is inconsistent. Thanks.