Why equation 22 is subtracting the current rather than adding?
simulate this circuit &ndash; Schematic created using CircuitLab

This is how your circuit looks like.

Look at \$i_c\$. I assumed it in that direction. If \$V_c\$ is the upper node of the capacitor, then it means \$i_c = C\frac{dV_c}{dt}\$. 

If you assumed \$i_c\$ to be in the opposite direction, then take \$i_c = -C\frac{dV_c}{dt}\$

Either way, you will end up in same answer.

In the solution, They have assumed the current directions like in the circuit I have drawn, so \$i_c = C\frac{dV_c}{dt}\$ 

Therefore \$i = 1.5i - i_c \$

Therefore \$v_c = (R1+R2).(1.5i - i_c) = (R1+R2).(1.5i - C\frac{dV_c}{dt})\$