# Laplace Transform

#### lkwah86

Joined Mar 9, 2007
22
Find the response y(t) of an LITC system described by the equation
[(d^2)y/(dt^2)] + 5[(dy)/(dt)] + 6y(t) = [(df)/(dt)] + f(t)
if the input f(t) = 3e^(-5t).u(t) and all the initial conditions are zero;
that is, the system is in zero-state.

i duno right hand side how 2 do...

#### DrNick

Joined Dec 13, 2006
110
first of all take the laplace transform of f(t). this would be

F(s) = 3/(s + 5)

now take the laplace transform of your ODE

remember derivatives are just multiplications by s (when there are no initial conditions), so the ODE becomes

s^2*Y(s) + 5*s*Y(s) + 6Y(s) = 3*s/(s+5) + 3/(s+5).

note that df/dt => s*(F(s))
d^2y/dt^2 => s^2Y(s)
etc...

Now solve for Y(s)
=>
Y(s) = [ 3*s/(s+5) + 3/(s+5) ] * 1/(s^2+5s+6)

now all you need to do is perform a partial fraction expantion on this, and then inverse transform!!

#### lkwah86

Joined Mar 9, 2007
22
but the f(t)=3e^(-5t)u(t)....
behind got u(t)...