Final answer:
Forming a differential equation in state variable format requires integrating the known derivatives and utilizing initial conditions to find constants. The provided information suggests a relation between v, u, and c, but additional context is required for precise initial conditions.
Step-by-step explanation:
To express the differential equation with input u in state variable format, we need to identify the knowns and unknowns and then form an equation that models the relationship between them. In the provided fragments, we recognize that the variable v is expressed as a fraction of c, and u' (the derivative of u) is also a multiple of c. Using this information, we can express u in terms of c.
Given that v = 0.500c and u' = -0.750c, we can integrate u' with respect to time to find u. However, the initial condition for the state variable φ(0⁻) = 0 needs to be taken into account when integrating to determine the constant of integration.
Assuming that u' represents the derivative of u with respect to time and v is a state variable related to u, the differential equation in state variable format might look something like:
u' = -0.750c and
v = 0.500c.
The initial condition is given as φ(0⁻) = 0. This can be translated to the initial condition for u if u and φ are related. Without additional context, however, it is challenging to provide the exact initial conditions for the state variables in this case.