Calculating an R-RC circuit
There's always more than one approach to solving a circuit problem but the approach I generally find most useful in this type of problem is to find the Thevenin equivalent resistance \$R_{TH}\$ "seen" by the capacitor.  This will allow you to find the time constant, \$\tau = R_{TH}C \$.

To find the Thevenin resistance, remove the capacitor and zero the voltage source (replace with wire).  Now, find the resistance between the terminals where the capacitor connects; that resistance is \$R_{TH}\$

If you've already found the voltage across the capacitor at t = 0 and t = \$\infty\$, just "connect them together" with the exponential function:

\$v_C(t) = [v_C(\infty) - v_C(0)](1 - e^{t/\tau}) + v_C(0)\$

For \$v_C(0) = 0\$, this simplifies to:

\$v_C(t) = v_C(\infty)(1 - e^{t/\tau})\$

Now that you have \$ v_C(t)\$, you have \$v_2(t) = V_1 - v_C(t) \$