# Finding frequency respone of a differential/integral LTI system

So suppose that we have an LTI system defined by the differential/integral equation below, where $x(t)$ and $y(t)$ denote the system input and output, respectively. How would I find the frequency response of this systems?

I haven't been taught this yet and the text book has no examples on it, I'm really confused. I'd really appreciate the help. This is the system:

$$\frac{d}{dt}y(t) + 2y(t) + \int_{-\infty}^t 3y(\tau)d\tau + 5\frac{d}{dt}x(t)-x(t) = 0$$

Frequency response deals with the steady-state behavior. Differentiating with respect to $t$ yields:

$$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+3y(t)=\frac{dx}{dt}-5\frac{d^2x}{dt^2}$$

The Laplace transform is $$(s^2+2s+3)Y(s)=(s-5s^2)X(s)$$ and we have

$$H(s)=\frac{Y(s)}{X(s)}=\frac{s-5s^2}{s^2+2s+3}$$ I think you can easily do the rest.

• I was able to figure it out thanks to this. I really appreciate your help. Thank you – Drew Nov 4 '16 at 5:05
• I was about to ask shouldn't that be 5s^2thanks for the edit – Drew Nov 4 '16 at 6:17
• Yes, sorry for the typo. – msm Nov 4 '16 at 6:21