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...
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!!