5E (Methods)

Use the causal Green function: it vanishes before the forcing time

t^{'}

. For

t > t^{'}

it solves the homogeneous equation, is continuous at

t = t^{'}

with value zero, and has first-derivative jump one. These conditions give

G (t, t^{'}) = {sinh (k (t - t^{'})) / k, 0, t \geq t^{'}, t < t^{'} .

Indeed the right derivative at the forcing time is one and the left derivative is zero, so

(\partial_{t}^{2} - k^{2}) G = δ (t - t^{'})

distributionally. If

k = 0

, use the continuous limit

G (t, t^{'}) = (t - t^{'})

on its causal support.

For

k \neq = 0

, differentiating the integral solution gives

\overset{x}{˙} (t) = \int_{0}^{t} cosh (k (t - t^{'})) f (t^{'}) d t^{'}, \overset{x}{¨} (t) = f (t) + k^{2} \int_{0}^{t} \frac{s i n h ( k ( t - t ^{'} ))}{k} f (t^{'}) d t^{'} .

Both initial values vanish and

\overset{x}{¨} - k^{2} x = f

. This verifies the requested representation and its sign. The

k = 0

formula similarly gives

\overset{x}{¨} = f

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