Find the inverse Laplace transforms for the given functions.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the Laplace Transform problems:

1. Find $L^{-1}\left[\frac{1}{s^3}\right]$

Use the inverse Laplace transform formula $L^{-1}\left[\frac{n!}{s^{n+1}}\right] = t^n$ .
Here, $n+1=3$ , so $n=2$ . We need $2!$ in the numerator.
$L^{-1}\left[\frac{1}{s^3}\right] = L^{-1}\left[\frac{1}{2!} \cdot \frac{2!}{s^3}\right]$
Apply the formula.
$L^{-1}\left[\frac{1}{s^3}\right] = \frac{1}{2} L^{-1}\left[\frac{2!}{s^3}\right] = \frac{1}{2} t^2$
The final answer is $\boxed{\frac{1}{2} t^2}$ .

2. Find $L^{-1}\left[\frac{1}{s^2(s^2+1)(s^2+9)}\right]$
Use partial fraction decomposition.
Let $X = s^2$ .
$\frac{1}{X(X+1)(X+9)} = \frac{A}{X} + \frac{B}{X+1} + \frac{C}{X+9}$
Solve for A, B, C.
For A, set $X=0$ : $A = \frac{1}{(0+1)(0+9)} = \frac{1}{9}$ .
For B, set $X=-1$ : $B = \frac{1}{(-1)(-1+9)} = \frac{1}{-8} = -\frac{

Answer