How many solutions does the ODE $u'=\frac{4tu}{u^2+t^2}$ have? Given the problem:
$$
\left\{
\begin{aligned}
u^\prime(t) & = \frac{4 t u(t)}{u^2(t)+t^2}  &&\text{ for } t\in(0,a)\\
u(0) & = 0  &&\text{ for } t=0
\end{aligned}
\right.
$$
Now I need to show that this ODE has infinitely many solutions. However, I cannot find any non-trivial. Has anyone a hint how to get a right solution.

Extending the hint by Semiclassical: 

If $u(t)=\pm t\sqrt{3}$, then $u'(t)=\pm \sqrt{3}=\dfrac{4t u(t)}{u(t)^2+t^2}$ and $u(0)=0$.  

For $b\in [-\sqrt{3},\sqrt{3}]$ consider the initial value problem $u(1)=b$ for this ODE. Let $u_b$ be its solution, which exists for $t>0$. Since for $t>0$ uniqueness holds, solution curves do not cross: therefore 
$$u_{-\sqrt{3}}(t)\le u_b(t)\le u_{ \sqrt{3}}(t),\quad t>0$$
and the conclusion follows.
