Tricky differential equation is solveable? $\frac{dy}{dt}=\sqrt{f(t)-y}$ It would be great to solve this problem! But I think maybe not possible because of the square root... is there anything that can be done with this? I guess this is invalid right:
$$\left(\dfrac{dy}{dt}\right)^2=f(t)-y$$
My model is self referencing (that is rate of change of $y$ is dependent on the difference between $z$ and $y$) but when $z=y$ , $\dfrac{dy}{dt}=0$ .
help... havent been to uni for a while and cant find example similar. I donth think integration factor will work... what about Laplace?
cheers 
 A: You will not find a close solution, except in some very particular cases such as $f$ is a constant. The uniqueness is not true in general for such a problem since $\sqrt{f(t)-y}$ is only Hölder continuous with exponent $1/2$. However, uniqueness holds say near $t=0, y=0$ provided $f(0)>0$, say with $f$ continuous:
$$
\frac{dy_j}{dt}=\sqrt{f(t)-y_j}, j=1,2, y_j(0)=z\Longrightarrow y_2(t)-y_1(t)=\int_0^t\bigl(\sqrt{f(s)-y_2(s)}-\sqrt{f(s)-y_1(s)} \bigr)ds
$$
which implies with $\rho(t)=\vert y_2(t)-y_1(t)\vert, t\ge 0$
$$
0=\rho(0)\le\rho(t)\le \int_0^t
\frac{\rho(s)}{\sqrt{f(s)-y_2(s)}+\sqrt{f(s)-y_1(s)}}ds.
$$
Since we may assume $f(0)-y_j(0)>0$ and by continuity that for some small enough positive $T$,
$$s\in(0,T),\quad
f(s)-y_j(s)\ge \frac{f(0)-y_j(0)}2,
$$
we get the differential inequality
$0=\rho(0)\le
\rho(t)\le C\int_0^t \rho(s) ds
$
which (Gronwall) implies $\rho\equiv 0$. 
A: Let $u=\sqrt{f(t)-y}$ ,
Then $y=f(t)-u^2$
$\dfrac{dy}{dt}=\dfrac{df(t)}{dt}-2u\dfrac{du}{dt}$
$\therefore\dfrac{df(t)}{dt}-2u\dfrac{du}{dt}=u$
$2u\dfrac{du}{dt}+u=\dfrac{df(t)}{dt}$
This belongs to an Abel equation of the second kind.
Let $u=-\dfrac{v}{2}$ ,
Then $\dfrac{du}{dt}=-\dfrac{1}{2}\dfrac{dv}{dt}$
$\therefore\dfrac{v}{2}\dfrac{dv}{dt}-\dfrac{v}{2}=\dfrac{df(t)}{dt}$
$v\dfrac{dv}{dt}-v=2\dfrac{df(t)}{dt}$
This belongs to an Abel equation of the second kind in the canonical form.
Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf or in http://www.iaeng.org/IJAM/issues_v43/issue_3/IJAM_43_3_01.pdf
