Final answer:
The steady-state response of a linear system to a sinusoidal input is also sinusoidal at the same frequency, with potentially different amplitude and phase. The specific amplitude and phase shift will depend on the system's characteristics and initial conditions, which are not provided.
Step-by-step explanation:
The student has presented a differential equation y(t) = x(t) and specified an input signal x(t) = 2 sin(t)u(t). To find the steady-state response of this system, we need to consider what behavior the system has after any transients have died out when it is driven by a periodic input signal. In the context of linear systems, which the given equation implies, the steady-state response to a sinusoidal input will also be sinusoidal at the same frequency but potentially with different amplitude and phase. Here, the input signal is already given in a sinusoidal form with a unit step function u(t) indicating that the input is applied at t ≥ 0.
In a typical solution to a differential equation with a sinusoidal driving force, we would expect the steady-state response to take the form y(t) = A sin(t + φ)u(t) or y(t) = A cos(t + φ)u(t) where A is the amplitude and φ is the phase shift brought about by the system's response characteristics. To provide a more specific answer, further information about the system's properties and initial conditions would be required.