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I have a major question. Please take a look.

I have this differential equation (DE): $$ \frac{d^2y(t)}{dt} +\frac{dy(t)}{dt} +4y(t)= \frac{dx(t)}{dt} +2x(t) $$ And I have to find impulse response (IR) and frequency response (FR).

We can "translate" the given DE into:

$$ Y(\omega)(j\omega)^2 + Y(\omega)5(j\omega) + 4Y(\omega)= j\omega X(\omega)+2X(\omega) $$

So i am reaching a point where $H(\omega) = Y(\omega)/X(\omega) = (j\omega +2)/[(j\omega)^2 +5j\omega + 4]$ .

The my main problem is that i dont know how to continue from this point, because in the numerator is that $j\omega$, otherwise i have my standard methodology to solve it.

So what am I supposed to do for this point ?

Thanks

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  • 4
    $\begingroup$ Hint: If $x(t)$ has transform $X(\omega)$, what is the transform of $y(t) = \frac{\mathrm d}{\mathrm dt}x(t)$? If my transform table has $x(t) \leftrightarrow X(\omega)$ listed, can I deduce the inverse Fourier transform of $j\omega X(\omega)$ from this information? $\endgroup$ – Dilip Sarwate Nov 18 '12 at 3:52
  • $\begingroup$ Your conversion of the DE seems to indicate the first derivative of y(t) in your original DE was intended to be multiplied by five. Is this the case? This would be the difference between real and complex poles for the system you describe. $\endgroup$ – Eric Nov 26 '12 at 16:19
  • $\begingroup$ The two equations you have written down do not match, therefore it's impossible to answer your question, as you have not followed up on which one is correct. $\endgroup$ – Phonon Oct 26 '13 at 5:08