Solving Vout for a differentiating op-amp with added resistor?
You can solve the current circuit using Laplace transformations. Assuming node 3 is at virtual ground.
$$ \frac{V_s}{R_1 + \frac{1}{sc}} = -\frac{V_o}{R_2}$$

$$V_o = \frac{R_2V_ssC}{R_1sC+1}------(1)$$

Now you can solve this one easily by using inverse laplace tranformation and reach at this solution,

$$ V_o =  \frac {R_2}{R_1}V_s-\frac{1}{R_1C}e^\frac{-t}{R_1C}\int(V_s.dt)  $$