I was hoping I could get some help checking my work through.
$\mathcal{L}\{x'(t)\} = sX(s)-1$ and $\mathcal{L}\{x''(t)\} = s^2X(s)-s$
Using the equality $\mathcal{L}\{-tf(t)\} = F'(s)$ we can rewrite the original system as:
$-\frac{d}{ds}[s^2X(s)-s]+[sX(s)-1]-\frac{d}{ds}[X(s)] = 0$
$(s^2+1)X'(s)+sX(s)=0$
$\frac{X'(s)}{X(s)} = -\frac{s}{s^2+1}$
Now I'm not sure where to go from here. How can I make this differentiation easier to understand so I can solve the problem? My textbook says the answers should be:
$X(s) = \frac{C}{\sqrt{s^2+1}}$
but I don't see how I can get that from my above formulation. Any help?