I'm stuck with a simple integration problem. Haven't done this in a while so I've gone a little rusty. Little help would be appreciated. Well anyway, here goes:
$dy/dt=\sqrt{{u}{y}+{v^2}}$
My attempt:
for the sake of simplicity let $dy/ty = T$
$T^2=uy+v^2$
$T^2dy/dt=(uy+v^2dy)/dt$
$T^2dy=uy+v^2dy$
$T^3/3+C=u^2/2*y^2/2+v^3/3$
$T^3/3=u^2/2*y^2/2+v3/3-C$
And I'm stuck. Quite frankly I'm not even sure that the two last steps were right.
As suggested by user88595, the differential equation is separable since you can write (as user88595 wrote) $$\int\dfrac{dy}{\sqrt{uy + v^2}} = \int dt = t$$ in which you can recognize a quite classical integral (if you do not, use a change of variable such that $uy=v^2z$. Doing so, you should arrive to $$\frac{2 \sqrt{u y+v^2}}{u}=t+C$$ which, in turn, leads to $$y=\frac{t^2 u^2+2 C t u^2+C^2 u^2-4 v^2}{4 u}$$