Non-linear differential equation

That's a common way to do it and you're left with:

[tex]u^2=4k\sqrt{v}+c[/tex]

or:

[tex]\left(\frac{dv}{dt}\right)^2=4k\sqrt{v}+c[/tex]

taking square roots, separate variable again and get:

[tex]\frac{dv}{\sqrt{4k\sqrt{v}+c}}=\pm dt[/tex]

That however looks kinda' messy to integrate although Mathematica gives a nice expression for the left side but then you get an implicit expression for v in terms of t:

[tex]g(v)=\pm(t+c_2)[/tex]

which is not solvable explicitly for v(t). Doesn't look like it anyway.