# The impulse response, $h(t)$, of a system

I have this differential equation $$y''(t) + y'(t) + y(t) = x(t) - 3x'(t)$$

I found the system response of the system:

$$H(f) = \frac{1 - 6\pi fj}{(2\pi fj)^2 +2\pi fj+1}$$ which is equivalent to

$$H(s) = \frac{1 - 3s}{s^2 +s+1}$$

My question is: How to find the impulse response? If I factorize the denominator, it's not gonna be pretty. e.g., $s + \frac{1 +\sqrt{-3}}{2}$

Also, what does the following mean?

• Find the output for $t \ge 0$ with initial conditions $y(0) = 1$, $y'(0) = -1$ when the input is $2cos(6t)u(t)$. Identify the zero state, the zero input, the transient, and the steady state responses.
• Consider reviewing material on partial fraction expansion and inverse Laplace transforms. – porten Jan 3 '14 at 3:22
• i'm still learning what's appropriate and what's not at SE. this surely looks like homework. is there a reason that asking the prof or the TA is not possible? – robert bristow-johnson Jan 3 '14 at 3:44
• that said, to find $h(t)$, you do have to split $H(s)$ into partial fractions which means you do have to factor the denominator which means complex conjugate roots. – robert bristow-johnson Jan 3 '14 at 3:46
• it's the standard stuff from a 2nd or 3rd year EE class. this has an answer, but i am loathe to just do it for you. but i'll help you do it, step-by-step. so, do you know how to do partial fraction expansion? – robert bristow-johnson Jan 3 '14 at 4:20
• There is a self-study tag that can be used by those who claim that they are not doing homework problems when in fact they are solving problems that are like typical homework problems, such as the ones at the end of the chapter in a textbook. – Dilip Sarwate Jan 3 '14 at 11:18

firstly do partial fraction for H(s) and then use transform tables of Laplace Transform. Using inverse laplace and tables you must be able to find impulse response. I think you can use j^2 instead of -1 and find complex roots for denominator. Then, using transform pairs you will find a complex solution.