# Inverse laplace transforms

### Inverse laplace transforms

#### Lessons

To solve a differential equation using a Laplace Transform it will also be necessary to know how to use an Inverse Laplace Transform

Finding the Inverse Laplace Transform involves turning the Laplace Transform of a function back into that function.

$L^{-1}{F(s)}$ = $f(t)$

For example from the section Calculating Laplace Transforms we saw,

$L${3$e^{3t}$} = $\frac{3}{s - 3}$

So to find out the inverse of the Laplace Transform of $\frac{3}{s - 3}$:

$L^{-1}${$\frac{3}{s-3}$} = 3$e^{3t}$

The same rule of Linearity applies:

$L^{-1}${$a F(s)$ + $bG(s)$} = $aL^{-1}${$F(s)$} + $bL^{-1}$ {$G(s)$}

• 1.
What are inverse Laplace Transforms?

• 2.
Inverse Laplace Transforms

Find the inverse Laplace Transform of the following functions:

a)
$F(s)$ = $\frac{12}{s^{4}}$ - $\frac{4}{s - 6}$ + $\frac{3}{s}$

b)
$G(s)$ = $\frac{4}{3s^{2} - 12}$ - $\frac{3}{(s - 5)^{3}}$

• 3.
Calculate the inverse Laplace Transform of the following functions:
a)
$G(s)$ = $\frac{7s - 3}{s^{2} - 2}$

b)
$H(s)$ = $\frac{3s - 5}{s^{2} - 4s + 7}$

• 4.
For each of the following functions calculate their inverse Laplace Transform:
a)
$F(s)$ = $\frac{7s + 3}{(s - 3)(s + 7)}$

b)
$G(s)$ = $\frac{3 s^{2} + 4s - 7}{(s^{2} + 7)(s - 3)}$