In state-space form (§1.3.7) ,8.7a general class of th-order Ordinary Differential Equations (ODE), can be written as
denotes the time derivative of , and is a vector (any length) of the system input signals, if any. Thus, Eq.(7.9) says simply that the time-derivative of the state vector is some function depending on time , the current state , and the current input signals . The basic problem is to solve for the state trajectory given its initial condition , the system definition function , and the input signals for all .
In the linear, time-invariant (LTI) case, Eq.(7.9) can be expressed in the usual state-space form for LTI continuous-time systems: