Exists a Laplace transform property to calculate functions of the type below int_a^x f(x−y)g(y)dy, when a>0?

undergoe8m 2022-11-18 Answered
Laplace transform property for convolutions of the type
0 x f ( x y ) g ( y ) d y
is very well known. There exists a Laplace transform property to calculate functions of the type below
a x f ( x y ) g ( y ) d y ,
when a>0?
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

iletsa2ym
Answered 2022-11-19 Author has 22 answers
You should look at the bilateral Laplace transform :
L [ f ( t ) 1 t > 0 ] ( s )   L [ g ( t ) 1 t > a ] ( s ) = ( f ( t ) 1 t > 0 e s t d t ) ( g ( t 2 ) 1 t 2 > a e s t 2 d t 2 )
= ( 0 f ( t ) e s t d t ) ( a g ( t 2 ) e s t 2 d t 2 ) = a h ( τ ) e s τ d τ
where
h ( τ ) = ( f ( t ) 1 t > 0 ) ( g ( t ) 1 t > a ) ( τ ) = f ( τ t ) 1 τ t > 0 g ( t ) 1 t > a d t = a τ f ( τ t ) g ( t ) d t
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-09-09
Laplace transform using 2nd shifting Theorem. L ( U ( t 5 ) e t 5 )
a = 5, Therefore, f ( t 5 ) = e t 5
since L ( f ( t ) ) = F ( s )
f ( t ) = e t
and L ( f ( t ) ) = 1 s 1
Therefore the answer is e 5 s 1 s 1
It`s correct answer?
asked 2021-09-10
Find the Laplace Transform of the function.
f(x)=sinh(ax)
asked 2022-01-20
The author of my textbook asks to verify that the function:
y=23ln(1+x2)+C
solves the differential equation
dydx=x3y+yx3
However, this is an error and this y does not solve the differential equation. Is there a simple typo that makes the problem workable?
asked 2022-09-22
Calculate L 1 { 1 s 2 ( s 2 + a 2 ) } using the convolution theorem L { f g } = L { f } L { g }
My approach:
f ( t ) = L 1 { 1 s 2 } = t
g ( t ) = L 1 { 1 s 2 + a 2 } = 1 a sin ( a t )
L 1 { 1 s 2 ( s 2 + a 2 ) } = f g = f ( t τ ) g ( τ ) d τ = 1 a ( t τ ) sin ( a τ ) d τ
but the last integral is clearly divergent.
asked 2022-11-08
Why
L ( x n ) ( s ) = n s L ( x n 1 ) ( s )
This one line has got me stuck! I know that
L ( f ( x ) g ( x ) ) F ( s ) G ( s )
so how could this line be valid.Please help.
asked 2021-09-06
Using only the Laplace transform table, obtain the Laplace transforms of the following function:
cosh(4t)4t
where ”cosh” stands for hyperbolic cosine and cosh x=exex2
asked 2022-09-26
Please help me find the Inverse Laplace transform of:
F ( s ) = 1 s ( s 2 + 8 s + 4 )
After completing the square, I obtained
F ( s ) = 1 s ( ( s + 4 ) 2 12 )