求微分方程

的特解，已知当 $t = 0$ 时，$x = \frac{\mathrm{d}x}{\mathrm{d}t} = 0$。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先由辅助方程求互补函数，特解试作 $ke^{-t}$。将初始位置和初始速度分别代入，求两个常数。

The auxiliary equation $6m^2+3m+6=0$ gives $m=-\frac14\pm\frac{\sqrt{15}}4\mathrm{i}$. For a particular integral, take $x=ke^{-t}$, so $x'=-ke^{-t}$ and $x''=ke^{-t}$. Substitution gives $9ke^{-t}=e^{-t}$, so $k=\frac19$. The general solution is

Put $\omega=\sqrt{15}/4$. Differentiating the general solution gives

The initial conditions give

The first equation gives $A=-1/9$. Substituting this into the second gives $\omega B=1/12$, hence $B=1/(3\sqrt{15})$. Thus the required solution is