1. Step Functions

Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

The given function:

$\displaystyle f(t)=\left\{ \begin{array}{l l} 0,& t<2\\ (t-2)^3, & t \geq 2 \end{array}\right.$

Phew! Trying to create that in LaTeX was probably just as difficult

2. Originally Posted by TheBerkeleyBoss
Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

The given function:

$\displaystyle f(t)=\left\{ \begin{array}{l l} 0,& t<2\\ (t-2)^3, & t \geq 2 \end{array}\right.$

Phew! Trying to create that in LaTeX was probably just as difficult

Well, by the definition of "Laplace Transform" this is $\displaystyle \int_2^\infty e^{-st}(t- 2)^3 dt$. That should be doable with integration by parts- three times.

3. You can solve this by redefining the function in terms of the unit step function $\displaystyle u(t)$

$\displaystyle u2(t)=0$ $\displaystyle t<2$
$\displaystyle u2(t)=1$ $\displaystyle t>=2$

So the given function may be written as

$\displaystyle f(t)=u2(t)(t-2)^3$

Now, an important property regarding laplace transforms of unit step functions is

$\displaystyle L(ua(t)g(t-a)) = e^{-as}L(g(t))$

Therefore here, $\displaystyle g(t)=t^3$ and $\displaystyle a=2$

So the laplace transform can be evaluated using the above property