inverse laplace 1/(s*exp(3*s)) — step-by-step solution

The factor

e^{-3 s}

delays the original function by

3

: invert

\frac{1}{s}

, then replace

t

with

t - 3

Find the causal inverse transform, for t ≥ 0. By linearity, invert each partial fraction

The inverse Laplace formula for

\frac{1}{{(s - a)}^{n}}

t^{n - 1} \cdot \frac{\exp (a t)}{(n - 1)!}

; it gives for

\frac{1}{s}

\frac{\exp (0 \cdot t)}{0!}

⇥ⓘ A product with a zero factor is zero:

0 \cdot t

0

■Any nonzero number raised to the power 0 is 1:

\exp (0)

1

■By convention, the factorial of 0 is 1:

0!

1

⇥

\frac{1}{1}

⇥

1

Delayed by

3

, the original function is zero for

t < 3

: replace

t

with

t - 3

and multiply by the unit step

u (t - 3)

, which is 0 for

t < 3

and 1 for

t \geq 3

▶Inverse Laplace transform:

u (t - 3)