Transform the differential equation to find a nontrivial solution such that x(0) = 0.

$\displaystyle tx''+(t-2)x'+x=0 $

What I have so far after the laplace transform is:

$\displaystyle -2sX-s^2X'-X'-2X-2sX+X=0$

$\displaystyle \frac {X'}{X} = - \frac {4s +1}{s^2+1} $

Then taking the integral, I have $\displaystyle lnX = -2ln(s^2+1) - tan^{-1}s $