properties of laplace transform

Question

Obtain the transfer function for the following differential equation and check whether the input free solution is stable or not,

$$\frac{dx}{dt} + 3x = f(t)$$

Please help, I don't even know where to start.

Answer 1

Just solve it in Laplace: $$\dot{x}+3x=f(t)$$ Applying Laplace transform and derivative property $\scr{L}$$\{\dot{f}(t)\}=sF(s)-f(0)$, considering initial condition $f(0)=0$ : $$sX(s)+3X(s)=F(s)$$ $$X(s)(s+3)=F(s)$$ $$Y(s)=\frac{X(s)}{F(s)}=\frac{1}{s+3}$$

Using the transform $\scr{L}$$\{\frac{1}{s+a}\}=e^{-at}$ $$y(t)=e^{-3t}$$

Finally, you can see that it's stable. In fact the only pole of the function ($s=-3$) is negative. This is clear on the time axis too, because the transient response goes to zero as $t$ goes to infinity.