I am having difficulty finding an approach for the following problem:
Problem: "Take an arbitrary $\alpha \in (0,1)$. Construct a non-zero solution for
$$\frac{dy}{dx} = |y|^{\alpha}, y(0) = 0$$
and then show that there are infinitely many solutions.
Hint: First find a solution in the following form:
\begin{equation*} y(x) = \begin{cases} 0, \quad &\text{if } x \leq 0\\ Cx^\beta &\text{otherwise} \end{cases} \end{equation*}
with C and $\beta > 0$ to be determined."
My attempt: I solved the ODE for $y > 0$ and then $y < 0$, but didn't know what to do from there.
\begin{align} &\text{When } y > 0 \Rightarrow y(x) = ((1 - \alpha)x)^{-(1-\alpha)}\\ &\text{When } y < 0 \Rightarrow y(x) = -((1 - \alpha)x)^{-(1-\alpha)} \end{align}
But I don't know where to go from here, or if this is the right first step at all. Any help would be appreciated. Thank you!
If you plug $y=C x^{\beta}$ in the ODE we obtain conditions on $C$ and $\beta$. We get $$ C\beta x^{\beta-1} = C^\alpha x^{\alpha \beta}. $$ Hence we need $\beta-1=\alpha \beta$ and $C=C^\alpha$, i.e. $C\equiv 1$ (otherwise we get the trivial solution) and $\beta=\frac{1}{1-\alpha}\in (1,\infty)$.