Laplace transforms of functions teᵗ

(i) L{t eᵗ}

Use L{t f(t)} = −dF/ds with F(s) = L{eᵗ} = 1/(s − 1):

L{t eᵗ} = −d/ds [1/(s − 1)] = 1/(s − 1)²      (s > 1)

(ii) L{sinh at · sin at}

Write sinh at = (e^(at) − e^(−at))/2 and use L{e^(±at) sin at} by the first shifting theorem:

L{e^(at) sin at}  = a/((s − a)² + a²)
L{e^(−at) sin at} = a/((s + a)² + a²)

L{sinh at sin at} = (a/2) [ 1/(s² − 2as + 2a²) − 1/(s² + 2as + 2a²) ]
                  = (a/2) · 4as / [(s² + 2a²)² − 4a²s²]
                  = 2a² s/(s⁴ + 4a⁴)

since (s² + 2a²)² − 4a²s² = s⁴ + 4a⁴.

Answers: (i) 1/(s − 1)²; (ii) 2a²s/(s⁴ + 4a⁴).