Thread: Solving a Mass-spring-damped system with Laplace

the equation is given, $\displaystyle x'' + 4x' + 4x = f(t)$

where f(t) = { t , 0 < t < 2 ; 0 , 2 < t }

I am having trouble getting the laplace of the $\displaystyle f(t)$ function.

I believe you would get L{f(t)} == L { [1 - H(t-2)]t }

Which would be L{t} - L{ t*H( t-2 ) }

Which I think would be $\displaystyle 1/u^2 - (e^-2u)/u^2$

Which Is$\displaystyle (1-e^-2u)/u^2$, but this is incorrect.

The solution is $\displaystyle 1/u^2 - (e^-2u)[2/s +1/u^2]$

is there anyone that could help me? Thank you in advance

Why You simply don't compute the integral...

$\displaystyle \mathcal{L}\{f(t)\} = \int_{0}^{2} t\cdot e^{-ut}\cdot dt$ (1)

... ?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$