Second order opamp circuit problem
It's just a typo. Its obvious from equation 2. They meant:
$$\frac{dv_1(0+)}{dt} = \frac{-v_0(0+)}{C_2R_2}=0$$
That's why all \$\frac{dv_1(0+)}{dt}\$ terms vanished from equation 1, on last simplification.