The part of a response that dies away after an initial condition or an input change. In a stable linear ordinary differential equation, the decaying homogeneous solution gives a transient response relative to a chosen particular solution. A homogeneous solution that grows or persists is not a decaying transient.
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Transient response refers to the behavior of a system as it reacts to a change in its input or initial conditions before reaching a steady state. In engineering, particularly in control systems and signal processing, the transient response is critical in analyzing how a system responds over time to inputs such as step functions, impulse functions, or other time-varying signals.