Step Functions

**TheBerkeleyBoss** · April 13th 2010, 12:25 AM

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:

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

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

Thanks for reading!

**HallsofIvy** · April 13th 2010, 12:54 AM

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:

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

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

Thanks for reading!

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

**bandedkrait** · April 13th 2010, 11:48 PM

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

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

So the given function may be written as

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

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

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

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

So the laplace transform can be evaluated using the above property

Thread: Step Functions

LinkBack

Thread Tools

Display

Step Functions

Similar Math Help Forum Discussions

Step Functions

[SOLVED] Sequences of Step Functions

A question about step functions

Step Functions

Step Functions

Search Tags