Solve the initial-value problem
$$
\begin{aligned}
y^{\prime} &=\alpha y-\beta y^{2}, \\
y(0) &=y_{0}
\end{aligned}
$$
where $\alpha$ and $\beta$ are small positive numbers. Show that
$$
\lim _{x \rightarrow \infty} \phi(x)=\left\{\begin{array}{ll}
\alpha / \beta & \text { for } y_{0}>0, \\
0 & \text { for } y_{0}=0 .
\end{array}\right.
$$
For $y_{0}<0, \phi$ is unbounded as $x$ approaches a certain value depending on $y_{0}$.