Example 2. Investigate the stability of the solution of the degenerate equation for the equation

$$
\varepsilon \frac{d x}{d t}=x\left(e^{x}-2\right)
$$
Solution. The degenerate equation $x\left(e^{x}-2\right)=0$ has two solutions

$$
\text { 1) } x=0, \quad 2) x=\ln 2 \text {. }
$$

We have

$$
\left.\frac{\partial f(t, x)}{\partial x}\right|_{x=0}=\left.\left(e^{x}-2+x e^{x}\right)\right|_{x=0}=-1
$$

so the solution $x=0$ is stable;

$$
\left.\frac{\partial f(t, x)}{\partial x}\right|_{x=\ln 2}=\left.\left(e^{x}-2+x e^{x}\right)\right|_{x=\ln 2}=2 \ln 2>0,
$$

so the solution $x=\ln 2$ of the degenerate equation is unstable (Fig. 51).