This is just to confirm @user21 result using Maple. Too small to post this as comment.

    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.

![Mathematica graphics](https://i.sstatic.net/zTaua.png)

Note that Maple does not have FEM solver, only FDM.