Finding the zero-state response
\$i=C\dfrac{dv_o}{dt}\$, and Ohm's Law gives: \$v_i - v_o=RC\dfrac{dv_o}{dt}\$

This can be solved by the integrating factor method, and zero state means zero initial conditions, therefore take \$v_o(0)=0\$

For higher order (>2) circuits you'll need the Laplace Transform.