laplace transform with heaviside step function
The cleanest way to handle such problems is to write $$ t u(t2) = (t2+2) u(t2) = (t2) u(t2) + 2 u(t2)$$ Now take Laplace transforms and use the time shift property you wrote.
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Daniel B.
Updated on August 01, 2022Comments

Daniel B. over 1 year
There are some similar questions around but they didn't help me much.
I've got a graph that I'm supposed to perform a laplace transform on. From $0<t<2$, it's a ramp function from $f(0)=0 to f(2) = 10$ (slope of 5) and then from 2 to 6 it's a 2.5 slope from $f(2) = 10$ to $f(6) = 0$.
Another way to solve this, I think, would be to take derivatives until it's in the form of impulse functions and then do the laplace using the transformation for a second derivative, but this is what I was doing first and I got stuck ...
I translated the intervals into $5tu(t) 7.5tu(t2) + 2.5tu(t6)$, where $u(t)$ is a heaviside step function.
I did this thinking I could use the time shift transform, but got stuck with the $t*u(t2)$ part. The time shift transform is $f(ta)u(ta) = e^{as}$, but in all but the first case, t is not shifted, only the step function is.
Would I have to use the integral definition of the transform from this point, or am I missing something?

Daniel B. over 9 yearsIf anybody knows how I can draw a graph in these questions, that'd be helpful :)

Daniel B. over 9 yearsAlso bear in mind I'm coming from a rudimentary crash course in these transforms from an engineering textbook, I haven't taken differential equations yet.


Daniel B. over 9 yearsI ... wow. Um what just happened? Oh, I see, distributive. Clever :D But ... what do I do with the 2u(t2)? I can't use the transform for u(t) on that can I?

user44197 over 9 yearsYes. $u(t) \to 1/s$ Hence $u(t2) \to e^{2s}/s$.

Daniel B. over 9 yearsIt'd be ... 2*integral(e^(st) from 2>infinity ... scribble scribble

user44197 over 9 yearsUse time shift property! see my comment above

Daniel B. over 9 yearsOh hey. You responded ... Oh right, because u(t) is a special case of u(ta) where a is zero, right? I don't see that on my little chart. I'm allowed a cheat sheet on my exam, I should put that on there.

user44197 over 9 yearsGood luck! Time for me to head to bed...it is 1:30 AM here