Mathematica disagrees with solution obtained by odelin

I tried in Maxima

(%i1) load(contrib_ode);
(eq_1) C:/PROGRA~1/MAXIMA~1.0/share/maxima/5.15.0/share/contrib/diffequations/contrib_ode.mac
(%i2) odelin(-'diff(y,x,2)+x^4*y=0,y,x);
...trying factor method
...solving 5 equations in 3 variables
...trying the Bessel solver
...solving 16 equations in 5 variables
...solving 16 equations in 5 variables
(eq_2) {bessel_j(-1/6,-(%i*x^3)/3)*sqrt(-x),bessel_y(-1/6,-(%i*x^3)/3)*sqrt(-x)}
(%i3) y=a*bessel_j(-1/6,-(%i*x^3)/3)*sqrt(-x)+b*bessel_y(-1/6,-(%i*x^3)/3)*sqrt(-x);
(eq_3) y=bessel_y(-1/6,-(%i*x^3)/3)*b*sqrt(-x)+bessel_j(-1/6,-(%i*x^3)/3)*a*sqrt(-x)
(%i4) ode_check(-'diff(y,x,2)+x^4*y=0,y=a*bessel_j(-1/6,-(%i*x^3)/3)*sqrt(-x)+b*bessel_y(-1/6,-(%i*x^3)/3)*sqrt(-x));(eq_4) 0

So therefore as a check I dumped this solution into Mathematica and it does not work.  In Mathematica when you plug in the solution in line eq_3 into the differential eq you do not get zero.  In fact you get

(in 1) f[x_] = a BesselJ[-1/6, -x^3/3] Sqrt[-x] + b BesselY[-1/6, -x^3/3] Sqrt[-x]

(in 2) FullSimplify[-D[f[x], {x, 2}] + x^4 f[x] == 0]

output 

Sqrt[-x] x (a BesselJ[-(1/6), -(x^3/3)] + b BesselY[-(1/6), -(x^3/3)]) == 0

which by plugging in different values for x gives left hand side values # 0

On the other hand if you plug in Mathematica's answer to the same differential equation (which is different) and substitute it back in you get True as the output of FullSumplify[].

Does anybody know what's going on here?

Rich