<s>[INST] Solutions of $y'=3x(y-1)^{\frac{1}{3}}$ where $y(0)=1$ So i am looking for the solutions of this differential equation and i am having problems. I need to find $9$ solutions but i can only find $3$. I separated the variables and got this equation $(y-1)^{\frac{2}{3}}=x^2$ well from that we get only two functions $y=1-x^3$ and $y=1+x^3$ and also $y=1$. Where are the $6$ solutions, could someone please explain? Thanks
 [/INST] What about of
\begin{align}
y_4(x)&=\begin{cases}1&x<0\\x^3+1&x\ge0\end{cases},&y_5(x)&=\begin{cases}1&x<0\\-x^3+1&x\ge 0\end{cases},&y_6(x)&=\begin{cases}x^3+1&x<0\\1&x\ge 0\end{cases}\\[5pt]
y_7(x)&=\begin{cases}-x^3+1&x<0\\1&x\ge0\end{cases},&y_8(x)&=\begin{cases}x^3+1&x<0\\-x^3+1&x\ge 0\end{cases},&y_9(x)&=\begin{cases}x^3+1&x<0\\-x^3+1&x\ge 0\end{cases}?
\end{align}
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