To verify ODE solution, use the functional method, as described in tutorial/DSolveSolutionVerification.html i.e. use y
and not y[x]
in the DSolve
command.
Sometimes, as in this case, you have to help it a little also using Simplify
as mentioned in the above link.

For your example:
ClearAll[y, x]
ode = y'[x] == x Sqrt[y[x]];
sol = DSolve[ode, y, x]
Simplify[eq /. sol,
Assumptions -> Element[{x, C[1]}, Reals] && x > 0 && C[1] > 0]

If you do not use the assumptions, then Mathematica does not know that the sqrt is real or complex. Here is without the assumptions:
ClearAll[y, x]
ode = y'[x] == x Sqrt[y[x]];
sol = DSolve[ode, y, x]
Simplify[eq /. sol]

btw, I always wished Mathematica has an odetest()
function as in Maple, as it makes it little easier. This is the same thing in Maple
ode := diff(y(x),x)= x*sqrt(y(x));
sol:=dsolve(ode,y(x));
odetest(sol,ode);
0
If the answer is zero, then the solution satisfies the ODE.
ps. I just saw your edit and you wanted to verify x^4/16
. Ok, in this case, you write this:
ode = y'[x] == x Sqrt[y[x]];
sol2 = y -> Function[{x}, x^4/16];
Simplify[eq /. sol2, Assumptions -> x > 0]

The idea is to use Function
, so that it will work with derivatives.
{}
button above the edit window. The edit window help button?
is also useful for learning how to format your questions and answers. You may also find this meta Q&A helpful. (Your post is not formatted so well at this point -- sorry. Don't format code as TeX. You want to make it easy to copy and paste back into Mathematica.) $\endgroup$ – Michael E2 Aug 26 '16 at 3:07