Hello...
I have a Diff. Eq. Midterm that is worth 50% of my grade on Monday so any help is appreciated.
I am trying to prove this Theorem here, which says the Laplace transform, which I will denote by L, is L[e<sup>at</sup>*f(t)] = F(s-a), and taking the inverse of both sides, we get e<sup>at</sup>*f(t) = L<sup>-1</sup>[F(s-a)]
Now I am trying to prove that F(s-a) = L[e<sup>at</sup>*f(t)].
Here is what I have so far:
F(s-a) = e<sup>-(s-a)t</sup>*f(t)dt (which is all inside an integral with limits from zero to infinity) = e<sup>-st</sup>[e<sup>at</sup>*f(t)]dt.
I got the first part by the definition of a Laplace transform which is given by F(s) = e<sup>-st</sup>*f(t) all inside an integral with limits from zero to infinity.
Somehow I need to arrive at L[e<sup>at</sup>*f(t)]. I tried to do integration by parts but I don't think that is going to work here...Any ideas on how to approach this?
I have a Diff. Eq. Midterm that is worth 50% of my grade on Monday so any help is appreciated.
I am trying to prove this Theorem here, which says the Laplace transform, which I will denote by L, is L[e<sup>at</sup>*f(t)] = F(s-a), and taking the inverse of both sides, we get e<sup>at</sup>*f(t) = L<sup>-1</sup>[F(s-a)]
Now I am trying to prove that F(s-a) = L[e<sup>at</sup>*f(t)].
Here is what I have so far:
F(s-a) = e<sup>-(s-a)t</sup>*f(t)dt (which is all inside an integral with limits from zero to infinity) = e<sup>-st</sup>[e<sup>at</sup>*f(t)]dt.
I got the first part by the definition of a Laplace transform which is given by F(s) = e<sup>-st</sup>*f(t) all inside an integral with limits from zero to infinity.
Somehow I need to arrive at L[e<sup>at</sup>*f(t)]. I tried to do integration by parts but I don't think that is going to work here...Any ideas on how to approach this?