Laplace Transform Question

[wcz]

Given X(s) = [s+1]/[s^2 + 5s + 7]

What is V(s) if
a) v(t) = x(3t-4) u(3t-4)

b) v(t) = d^2 x(t)/dt^2

The given answer are
a) V(s) = [s+3] / [s^2 + 15s + 63] e^-4s/3
b) V(s) = [13s+28] / [s^2 + 5s + 7]

I got different solutions for both.
Could anyone show me the way to get the answers?
Thanks.

 

[zorro]

Hi wcz,

a) v(t) = x[3(t-4/3)]*u[3(t-4/3)],
and provided that x(t)=0 for t<0, then x(t)=x(t)*u(t) and v(t) = x[3*(t-4/3)] .
We have:

a scaling in time by 3:
Let y(t)=x(3*t)
Y(s)=1/3 * X(s/3)

a translation in time by 4/3: v(t) = y(t-4/3)
V(s) = Y(s)*exp(-4*s/3);

This is the same as the given answer.

b) It should be Y(s) = s^2 * X(s), with a different result than the given answer.

Regards

Z

 

[wcz]

Thanks Z.
I'm having another problem in solving Inverse Laplace Transform.
Could anyone show me the way to solve the Inverse laplace transform of the following Q,

X(s) = [ s^2 - 2s + 1 ] / [ s^2 ( s^2 + 4) ]

 

[steve10]

Factorization gives:

-1/(2s)+1/(4s^2) +(s/2+3s/4)/(s^2+4)
and check the formulas of laplace transformation.

 

[amraldo]

steve10 said:

Factorization gives:

-1/(2s)+1/(4s^2) +(s/2+3s/4)/(s^2+4)
and check the formulas of laplace transformation.

& i agree with that but i think factorization will give

-1/2s +1/4s^2 + (s/2 + 3/4 )/(s^2+4)

 

[steve10]

Thanks, amraldo. That was a typo.

 

[suvendu]

you can take help of ziemer and tranter book and see laplace transform chapter.