Inverse Laplace Transform of a fractional F(s)

[Italo Campoli]

Homework Statement

[/B]
Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor

2. The attempt at a solution

tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as well for both of them but i would have to rework them to get them to look similar and that's the part were i get tangled arround, i would like to see a step by steph solution if possible i'd really like to understand this one

Its driving me crazy its the last one i need to complete a test. In my class we haven't got to convolution yet, but a friend told me that i should try that, id appreciate if you guys help me with that by any possible method.

 

[pasmith]

$3s + 8 = 3(s + 1) - 1$

 

[Italo Campoli]

Thanks fot the hint but

ill give it a shot, tho i'veen struggling with it all day :S

 

[Ray Vickson]

Homework Statement

[/B]
Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor

2. The attempt at a solution

tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as well for both of them but i would have to rework them to get them to look similar and that's the part were i get tangled arround, i would like to see a step by steph solution if possible i'd really like to understand this one

Its driving me crazy its the last one i need to complete a test. In my class we haven't got to convolution yet, but a friend told me that i should try that, id appreciate if you guys help me with that by any possible method.

If you let
$$g_1(s) = \frac{1}{s^2 + 2s + 10} ,$$
can you find
$$f_1(t) = \left({\cal L}^{-1} \, g_1(s) \right) (t)?$$
Do you know the relationship between the inverse transforms of $g_1(s)$ and $s g_1(s)$? Therefore, do you know the inverse transform $f_2(t)$ of $g_2(s) = (3s + 8) g_1(s)$? Finally, do you know how to get the inverse transform of $g_1(s) g_2(s)$ from the two inverses $f_1(t)$ and $f_2(t)$? (Even if you do not have time to complete the computations, giving the required formulas ought to earn you some decent marks.)
Do you know the relationship between the inverse transforms of $1/(s^2+2s+10)$ and $s/(s^2+2s+10)$? Can you thus find the inverse transform of $(3s+8)/(s^2+2s+10)$? Finally, if

 

[Italo Campoli]

If you let
$$g_1(s) = \frac{1}{s^2 + 2s + 10} ,$$
can you find
$$f_1(t) = \left({\cal L}^{-1} \, g_1(s) \right) (t)?$$
Do you know the relationship between the inverse transforms of $g_1(s)$ and $s g_1(s)$? Therefore, do you know the inverse transform $f_2(t)$ of $g_2(s) = (3s + 8) g_1(s)$? Finally, do you know how to get the inverse transform of $g_1(s) g_2(s)$ from the two inverses $f_1(t)$ and $f_2(t)$? (Even if you do not have time to complete the computations, giving the required formulas ought to earn you some decent marks.)
Do you know the relationship between the inverse transforms of $1/(s^2+2s+10)$ and $s/(s^2+2s+10)$? Can you thus find the inverse transform of $(3s+8)/(s^2+2s+10)$? Finally, if

i really apreciatte your answer, i understood a bit but not as much as i woulda loved, see my proffesor hasnt got to those properties you menthion but ill keep your answer in mind for later I am sure it will be of a lot of use!

Now using the idea @pasmith gave me, i got this

seems a bit easier, now , the one on the left looks almost the same as t.cos(wt) and maybe with some rework/rewrite here and there the one in the right like t.sin(wt) , please correct me if I am wrong, i been in this excersise alone since this morning i might be going crazy already, i would like some advice in simple terms how to proceed from here, thank you a lot @Ray Vickson

EDIT: lel with the rush i forgot the { } on both inverses, meh i think it is visually understandable hehe

 

[Ray Vickson]

i really apreciatte your answer, i understood a bit but not as much as i woulda loved, see my proffesor hasnt got to those properties you menthion but ill keep your answer in mind for later I am sure it will be of a lot of use!

Now using the idea @pasmith gave me, i got this

seems a bit easier, now , the one on the left looks almost the same as t.cos(wt) and maybe with some rework/rewrite here and there the one in the right like t.sin(wt) , please correct me if I am wrong, i been in this excersise alone since this morning i might be going crazy already, i would like some advice in simple terms how to proceed from here, thank you a lot @Ray Vickson

EDIT: lel with the rush i forgot the { } on both inverses, meh i think it is visually understandable hehe

Even if your professor has not yet given you some properties, are you not allowed to look them up in your textbook, or to find them on-line or in the library?

 

[Italo Campoli]

Even if your professor has not yet given you some properties, are you not allowed to look them up in your textbook, or to find them on-line or in the library?

yes but i can't seem to understand them quite right if i see them by the first time i would need to look like a few examples steph by steph and such like in a class, I am just a rookie here :( trust me ill give your idea to a friend of mine who is more ahead in the career than me, I am sure he will explain it to me in baby steps or at least try hehe