This is just to confirm @user21 result using Maple. I used bvp[middefer] (For BVP problem like this one) in Maple to get solution. I am not sure what this maps to in NDSolve now if any.
restart;
ode := diff(X(x),x$2)=X(x)^(3/2)/sqrt(x);
bc := X(0)=1,X(10)=0;
sol := dsolve([ode,bc],X(x),`numeric`,method=bvp[middefer],abserr=0.001,maxmesh=8192*4);
plots:-odeplot(sol, x=0..10);
Similar (but not exact) looking solution is generated.
Note that Maple does not have FEM solver, only FDM.