常微分方程：

齐次方程 $u''+u'+u=0$ 的特征方程

$$r^2+r+1=0\Rightarrow r=\frac{-1\pm i\sqrt3}{2}.$$

因此

$$u_h=e^{-x/2}\left(C_1\cos\frac{\sqrt3 x}{2}+C_2\sin\frac{\sqrt3 x}{2}\right).$$

取常数特解 $u_p=1$（代入可得 $1$）。

通解为

$$\boxed{u(x)=1+e^{-x/2}\left(C_1\cos\frac{\sqrt3 x}{2}+C_2\sin\frac{\sqrt3 x}{2}\right)}.$$

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英文解析

The characteristic equation of the homogeneous equation $u''+u'+u=0$ is
$$r^2+r+1=0\Rightarrow r=\frac{-1\pm i\sqrt3}{2}.$$
Therefore
$$u_h=e^{-x/2}\left(C_1\cos\frac{\sqrt3 x}{2}+C_2\sin\frac{\sqrt3 x}{2}\right).$$
Taking a constant particular solution $u_p=1$ (substituting yields $1$).

The general solution is
$$\boxed{u(x)=1+e^{-x/2}\left(C_1\cos\frac{\sqrt3 x}{2}+C_2\sin\frac{\sqrt3 x}{2}\right)}.$$