The state equation of a second-order linear system is given by $$\mathop x\limits^ \bullet \left( t \right) = Ax\left( t \right),x\left( 0 \right) = {x_0}.$$
For $${x_0} = \left[ {\matrix{ 1 \cr { - 1} \cr } } \right],x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} \cr { - {e^{ - t}}} \cr } } \right]$$ and for $${x_0} = \left[ {\matrix{ 0 \cr 1 \cr } } \right],x\left( t \right) = \left[ {\matrix{ {{e^{ - t}}} & { - {e^{ - 2t}}} \cr { - {e^{ - t}}} & { + 2{e^{ - 2t}}} \cr } } \right]$$ when $${x_0} = \left[ {\matrix{ 3 \cr 5 \cr } } \right],x\left( t \right)$$ is

An unforced linear time invariant (LTI) system is represented by $$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 1} & 0 \cr 0 & { - 2} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right].$$$

If the initial conditions are x₁(0)= 1 and x₂(0)=-1, the solution of the state equation is

Consider the state space system expressed by the signal flow diagram shown in the figure.

The corresponding system is

Consider the state space model of a system, as given below

The system is