## anonymous 5 years ago Find inverse Laplace transform by using convolution. Please step by step in integral part: 1/s^2+5s+6

1. anonymous

first factor s^2+5s+6 and post what you got

2. anonymous

I/(s+2)(s+3) so inverse laplace of those as F(s)=1/(s+2) and g(s)=1/s+3 and taking inverse we have F(s)=e^(-2t) and G(s)=e^(-3t)

3. anonymous

if you are taking inverse, it would be in time domain. if F(s) = 1/(s+2), then F(s) cannot also be equal to e^(-2t)

4. anonymous

if H(s) = 1/s^2+5s+6 and F(s)=1/(s+2) and G(s)=1/(s+3), then H(s) = F(s)G(s)

5. anonymous

according to convolution, product of two functions in the angular frequency domain can be represented as convolution of their corresponding time domain functions.

6. anonymous

so, h(t) = f(t)*g(t) = $\int\limits_{0}^{t}f(\tau)g(t-\tau) d \tau$

7. anonymous

if F(s) = 1/(s+2), then inverse laplace transform of it is f(t) = e^(-2t)

8. anonymous

you have all the information you need to solve this.

9. anonymous

so final answer after integrating would be 3^(-2t)-e^(-3t)

10. anonymous

I ment to say e^(-2t)-e^-3t

11. anonymous
12. anonymous

I dont know what the final answer is. I didn't actually try to solve it. I showed you the method. I trust you can take it from there

13. anonymous

thank you just checked inm solution and it is

14. anonymous

okay great!

15. anonymous

$H(s)=1/s(s^2+9)=A/s+Bs+C/(s^2+9) by \partial fraction I get (1/9s) -(s/9(s^2+9)$

16. anonymous

for you to use convolution, you have to express your H(s) as a product of two or more functions in s- domain

17. anonymous

you can't express them as a summation.

18. anonymous

$so \Im \left with 1/9 L^-1 (1/s) -1/9 L^-1(s/s^2+9)$

19. anonymous

so F(s)=1/9 and G(s)=-1/9cos3t

20. anonymous

$\int\limits_{0}^{t}1/9-1/9\cos3rhod \rho= ?$

21. anonymous

22. anonymous

i dont know what you are trying to ask. I cannot help you with actually finding the values to your problems. I can only show you how. Go through that video. It explains it well with an example similar to your problem

23. anonymous

cool