Laplace Transformations Step Functions

[Mikesgto]

Homework Statement

The attachment is the problem.

Homework Equations

The Attempt at a Solution

I understand how to go about solving the laplace transformations but I have no idea how to start with the Heaviside functions for the 5t and the 30. What I got was 5t+30U₆(t) but it turned out to be wrong. Any help would be greatly appreciated.

 

[vela]

That's close. If t=7, the forcing function should be equal to 30, but your expression would give 5(7)+30U₆(7) = 35+30 = 65. Can you see why it's wrong and how to fix it?

 

[Mikesgto]

5t-5U₆(t)(t-6)

I got this one after a ton of work, but I am failing to grasp the concept going behind this.

 

[vela]

You can look at it this way. You only want the 5t term to contribute between t=0 and t=6, so after t=6, you have to subtract out 5t to cancel the other 5t term:

F(t) = 5t U₀(t) + 30 U₆(t) - 5t U₆(t)

(Often, it's understood that the input is 0 for t<0, so you don't write the U₀(t) in explicitly.) When t>6, the first and last terms cancel, and you're just left with F(t)=30.

You can also look at it like this, which is the way I prefer. If you rearrange the terms slightly, you can write

F(t) = 5t (U₀(t) - U₆(t)) + 30 U₆(t)

The factor U₀(t)-U₆(t) is equal to 1 between t=0 and t=6 and is equal to 0 everywhere else, so it turns the 5t term on between t=0 and t=6 and disables it for the other times.

 

[Mikesgto]

ok, so the next problem i have is attached and the solution i came up with is
3t-3tU₉(t)

Am I on the right path there?

 

[vela]

Yup, that's right.

 

[Mikesgto]

Thanks so much for all the help. The next part of my solutions are finding out F(s). Do I treat the -3t attached to the U₉(t) as my f(t)?

 

[vela]

I'm not sure what you're asking. It might help if you show us your calculations.