I'd like to extend my comments to an answer. For those in _v12_ or higher, `FiniteElement` is a possible choice for this problem, as shown in [user21's answer](https://mathematica.stackexchange.com/a/219608/1871). But, if you're in a version lower than _v12_ but higher than _v9_, it becomes a bit more troublesome, because 

1. nonlinear `FiniteElement` isn't implemented yet.

2. `Shooting` method can't handle the problem well, which is [an arguable backslide](https://mathematica.stackexchange.com/a/130373/1871).

[![enter image description here][1]][1]

_As we can see, though `ndsz` warning is generated, `NDSolve` manages to find the desired result in v9._

OK, so what to do? Well, to be honest I don't know if the following solution will cause other problem in _v10.0_, because _v10.0_ is a quite unstable version, but it does work in _v9_ and _v12.1_:
    
    rin = 10^-30;
    
    psol = ParametricNDSolveValue[{D[χ[x], {x, 2}] == (χ[x]^(3/2))/Sqrt[x], χ'[
         10] == d, χ[10] == 0}, χ, {x, rin, 10}, d]
    
    drule = FindRoot[psol[d][rin] == 1, {d, 0 (* choose -1/10 if in v9 *)}] // Quiet
    (* {d -> -0.0116574} *)

    Plot[psol[d /. drule][r] // Evaluate, {r, rin, 10}, PlotRange -> All]

Alternatively, we can turn to finite difference method (FDM). I'll use [`pdetoae`](http://mathematica.stackexchange.com/a/127997/1871) for the generation of finite difference equations:

    rin = 0;
    
    eq = D[χ[x], {x, 2}] Sqrt[x] == (χ[x]^(3/2));
    
    bc = {χ[rin] == 1, χ[10] == 0};
    
    points = 25; domain = {rin, 10}; grid = Array[# &, points, domain]; difforder = 2;
    
    del = #[[2 ;; -2]] &;
    (* Definition of pdetoae isn't included in this post,
       please find it in the link above. *)    
    ptoafunc = pdetoae[χ[x], grid, difforder];
    
    ae = ptoafunc@eq // del;
    
    initialguess[x_] = 0;
    
    solrule = FindRoot[{ae, bc}, Table[{χ[x], initialguess[x]}, {x, grid}]]
    
    sol = ListInterpolation[solrule[[All, -1]], grid]


  [1]: https://i.sstatic.net/RwzBe.png