Initial Value Problem 
We have initial value problem $y''=x^{1/3}y$ with $y(0)=y'(0)=0$. 
  Does it have a unique solution? 

I have tried using the Picard-Lindelöf theorem, but I cannot reduce it to a order 1 ODE. Thank you.  

Check this

Theorem: Let $p(t), q(t)$, and $g(t)$ be continuous on $[a,b]$, then the differential equation
$$y'' + p(t)y' + q(t)y  =  g(t),\quad           y(t_0)  =  y_0,\,     y'(t_0)  =  y'_0 $$
has a unique solution defined for all $t$ in $[a,b]$

