Laplace Transform

I'm having trouble with this question. Can anyone please guide me.

My Attempt :

= 1/(s-1) * L{t^2*e^-t}
= 1/(s-1) * (2/(s+1)^3)
= 2/((s-1)(s+1)^3))

but that's not the answer , its 2/((s-1) s^3) somehow.
The question is attached below.

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Ray Vickson
Homework Helper
Dearly Missed
I'm having trouble with this question. Can anyone please guide me.

My Attempt :

= 1/(s-1) * L{t^2*e^-t}
= 1/(s-1) * (2/(s+1)^3)
= 2/((s-1)(s+1)^3))

but that's not the answer , its 2/((s-1) s^3) somehow
You are correct: it really is 2/[(s-1)(s+1)^3].

But on the test it says its 2/(s^3 *(s-1)), and wolfram alpha confirms it too... The question asked to find F(2).. according to my solution i will get 2/(3^3) which is 2/27 but the answer is 1/4

Ray Vickson
Homework Helper
Dearly Missed
But on the test it says its 2/(s^3 *(s-1)), and wolfram alpha confirms it too... The question asked to find F(2).. according to my solution i will get 2/(3^3) which is 2/27 but the answer is 1/4
If '*' means multiplication, then your answer is correct (and Maple confirms this). If '*' means 'convolution' then you answer is wrong.

In my answer which is 2/((s-1)*(s+1)^3)) , '*' does mean multiplication, so F(2) should be 2/27 , is that correct? ... test got 1/4 however.

Ray Vickson
Homework Helper
Dearly Missed
In my answer which is 2/((s-1)*(s+1)^3)) , '*' does mean multiplication, so F(2) should be 2/27 , is that correct? ... test got 1/4 however.
I agree with you; if F(s) means the product as given in the question you wrote, then F(2) = 2/27, NOT 1/4. Maybe you wrote the wrong question?

I can see no way to get the answer 2/[(s-1)s^3] using any familiar kinds of operations. A product does not give this, and a convolution also does not give this. You say that Wolfram Alpha gives this; well, I don't believe it. Maple does not give this, and it is essentially equivalent to Mathematica in strength and applicability, so is at least as powerful as Wolfram Alpha (which uses a subset of Mathematica routines). If you say Wolfram Alpha gets this you will need to present exactly what commands you used; perhaps you were solving a different problem without realizing it.

This is the exact question... no option of 2/27 :(

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vela
Staff Emeritus
Homework Helper
I'm having trouble with this question. Can anyone please guide me.

My Attempt :

= 1/(s-1) * L{t^2*e^-t}
= 1/(s-1) * (2/(s+1)^3)
= 2/((s-1)(s+1)^3))

but that's not the answer , its 2/((s-1) s^3) somehow
I don't get what you did there.

Pull the exponential in front back inside the integral and then identify the integral as a convolution of two functions. What two functions are they?

I'm confused here... are u saying i should put e^t inside integral and do L{e*t^2}?

vela
Staff Emeritus
Homework Helper
No, how did you get that? You do realize the integral isn't a Laplace transform, right? When you pull the exponential back in, you get
$$f(t) = \int_0^t \tau^2 e^{t-\tau}\,d\tau.$$ You should recognize the form of that integral.

oh wow , i get it now. Thanks you so much for helping.

Ray Vickson