How is this circuit providing solution of the differential equation d^2 v/dt^2 = -9v?
The output voltage v is the result of integrating twice. So whatever was before the integrating stages is d^2 v / dt^2 by definition. Since that node is driven strongly by the inverting gain stage, it also needs to be -9 v. Hence the differential equation from the figure caption.

Perhaps the connection is easier to make if you convert the differential equation into the equivalent integral equation by integrate both sides twice?