Space state circuit modeling question
You know that \$V_{R_1} = V_{C_2}\$
So replacing \$V_{R_1}\$ by \$V_{C_2}\$ in your equation
\$V_{C_1} + V_{R_1} - V_1 = 0\$
leads to
\$V_{C_1} + V_{C_2} = V_1\$
Differentiating results in
\$d(V_{C_1}) /dt + d(V_{C_2})/dt = 0\$
or
\$d(V_{C_1}) /dt = -d(V_{C_2})/dt\$
Inserting that in your last equation leaves you with only one derivative.