# Inverse Laplace : with Convolution

## Homework Statement

Use the convolution property to obtain the inverse Laplace transform of
F(s)= e-3s * ((3s+15)/s2+25)

## The Attempt at a Solution

= (3*(s/s2+52) + 15*(1/s2+52)) *e-3s

Using table of Laplace:
3*(s/s2+52) = 3*cos(5*t) = T7
15*(1/s2+52) = 15/5*sin(5*t) =T18
e-3s = s(t-a) = s(t-3) (T16)

=(3*cos(5*t)+15/5*sin(5*t))*s(t-3)

Convolution : int 0 to t (f(u)*g(t-u))du
integral 0 to t ((3*cos(5*u)+15/5*sin(5*u))*s((t-u)-3)) du

Something is wrong in my work here, and I cant find where. :([/SUP]

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Use the convolution property to obtain the inverse Laplace transform of
F(s)= e-3s * ((3s+15)/s2+25)

## The Attempt at a Solution

= (3*(s/s2+52) + 15*(1/s2+52)) *e-3s

Using table of Laplace:
3*(s/s2+52) = 3*cos(5*t) = T7
15*(1/s2+52) = 15/5*sin(5*t) =T18
e-3s = s(t-a) = s(t-3) (T16)

None of the above equalities are true; you have functions of s "equal" to functions of t.

=(3*cos(5*t)+15/5*sin(5*t))*s(t-3)

Convolution : int 0 to t (f(u)*g(t-u))du
integral 0 to t ((3*cos(5*u)+15/5*sin(5*u))*s((t-u)-3)) du

Something is wrong in my work here, and I cant find where. :([/SUP]
And you are apparently using s for two different things. With 156 posts, it is past time for you to learn to use latex. It isn't worth the effort to decipher what you have written.

vela
Staff Emeritus
Homework Helper

## Homework Statement

Use the convolution property to obtain the inverse Laplace transform of
F(s)= e-3s * ((3s+15)/s2+25)
Is the asterisk supposed to denote multiplication or convolution?

You need to use parentheses as appropriate. Assuming the asterisk is supposed to be multiplication, what you wrote means
$$F(s) = e^{-3s}\left(\frac 3s + \frac{15}{s^2} + 25\right),$$ which is quite different from what you intended.

Using table of Laplace:
3*(s/s2+52) = 3*cos(5*t) = T7
15*(1/s2+52) = 15/5*sin(5*t) =T18
e-3s = s(t-a) = s(t-3) (T16)

=(3*cos(5*t)+15/5*sin(5*t))*s(t-3)
Is there some reason you're not simplifying 15/5?

Convolution : int 0 to t (f(u)*g(t-u))du
integral 0 to t ((3*cos(5*u)+15/5*sin(5*u))*s((t-u)-3)) du

Something is wrong in my work here, and I can't find where. :([/SUP]
Why do you think something's wrong?

Besides the problems LCKurtz pointed out with what you've written, did you notice that when you wrote "=(3*cos(5*t)+15/5*sin(5*t))*s(t-3)," you never said what it was equal to and that you're using * to mean both multiplication and convolution? You should make an effort to be less sloppy, especially when you're trying to communicate with others.