# Step function question

• Apr 14th 2010, 08:43 AM
TheBerkeleyBoss
Step function question
I'm having difficulty figuring out what to do with this step function problem:

The instructions are to find the Laplace transform of the given function...

$f(t) = (t-3)u_2(t)-(t-2)u_3(t)$

I've looked at the rules for the transformations but I'm perplexed because it seems the $u_2$ should correspond with $(t-2)$ rather than $(t-3)$. This is my last problem from this section and unfortunately I'm stuck. I'm getting exponentials in my answer but it's not perfectly matching up with the given solution.

• Apr 14th 2010, 05:01 PM
TheEmptySet
Quote:

Originally Posted by TheBerkeleyBoss
I'm having difficulty figuring out what to do with this step function problem:

The instructions are to find the Laplace transform of the given function...

$f(t) = (t-3)u_2(t)-(t-2)u_3(t)$

I've looked at the rules for the transformations but I'm perplexed because it seems the $u_2$ should correspond with $(t-2)$ rather than $(t-3)$. This is my last problem from this section and unfortunately I'm stuck. I'm getting exponentials in my answer but it's not perfectly matching up with the given solution.

Here is the trick you need.

Note that

$(t-3)\mathcal{U}(t-2)=[(t-2)-1]\mathcal{U}(t-2)=(t-2)\mathcal{U}(t-2)-\mathcal{U}(t-2)$

Now these should be the form for you tables or note that

$\mathcal{L}(f(t-a)\mathcal{U}(t-a)e^{-as}\mathcal{L}(f(t))$

So using this on the first half gives

$\frac{e^{-2s}}{s^2}-\frac{e^{-2s}}{s}$

Now use a similar trick on the 2nd term.

I hope this helps