How many solutions does the ODE have? The question :
Given ODE :
$$
\begin{cases}
y'-a^2(y')^3-\frac{\sin(x)}{x+y}=0 \\
y(0)=1
\end{cases}
$$
Write how many solutions does the system have for $a=0$ and $a \ne 0$.
My try :
for $a=0$ we have a unique solution by Picard's theorem.
In the case where $a \ne 0$ I'm stuck and can't understand...

If $a\ne 0$, your first equation is a cubic equation in $y'$. So you have three roots:
$$y'_{1,2,3}=f_{1,2,3}(x,y(x))$$
For each root you have a unique solution (Picard–Lindelöf theorem). All you need to show is that your roots are different. Start from the cubic roots formula.
