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I was trying to understand the way this problem was solved and I got confused with the latter part of the solution. I encircled the part that confused me. They seem to contradict each other. If dv(0+)/dt = 0 why is it dv(0+)/dt = -1 in the other one? Please explain. TIA!

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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.

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