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Stability of h(t) = e^(4t)u(t)

Short answer

The system is unstable because its impulse response is not absolutely integrable: ∫0^∞ e^{4t} dt = ∞.

Test the stability of the system h(t) = e^(4t)·u(t).

An LTI system is BIBO stable if its impulse response is absolutely integrable: ∫|h(t)| dt < ∞ over (-∞, ∞).

∫|h(t)| dt = ∫₀^∞ e^(4t) dt = [e^(4t)/4]₀^∞ = ∞

The integral diverges because the exponent is positive (a pole in the right half of the s-plane at s = 4).

Answer: the system is unstable.

✓ Answer checked by Claude Opus 5.5 (high effort)

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