Consider the first-order ordinary differential equation

$$\frac{\mathrm{d} y}{\mathrm{~d} x}=f_{1}(x) y+f_{2}(x) y^{p}$$

where $y \geqslant 0$ and $p$ is a positive constant with $p \neq 1$. Let $u=y^{1-p}$. Show that $u$ satisfies

$$\frac{\mathrm{d} u}{\mathrm{~d} x}=(1-p)\left[f_{1}(x) u+f_{2}(x)\right]$$

Hence, find the general solution of equation $(*)$ when $f_{1}(x)=1, f_{2}(x)=x$.

Now consider the case $f_{1}(x)=1, f_{2}(x)=-\alpha^{2}$, where $\alpha$ is a non-zero constant. For $p>1$ find the two equilibrium points of equation $(*)$, and determine their stability. What happens when $0<p<1$ ?