A metal rod of length L having uniform cross section area A and variable linear mass density ${\lambda }_x={\lambda }_0(1+\frac{x}{L}),x$  is measured from the end A. Rod is placed on a smooth horizontal surface. A constant force  $F_0$  is applied at end B. Young’s modulus of rod is Y, the value of energy stored in the rod will be  $\frac{1}{x}\left(\frac{19F_0^{2}L}{15YA}\right)$, then x is _______