Duhamel's Integral

For a linear time‑invariant (LTI), causal system, Duhamel's integral expresses the time‑domain response y(t) to an arbitrary input f(t) as the convolution of the system's impulse response h(t) with the input: y(t) = ∫_0^t h(t − τ) f(τ) dτ (plus any homogeneous solution accounting for initial conditions).