求微分方程

满足当 $t=0$ 时，$x=\dfrac{\mathrm{d}x}{\mathrm{d}t}=0$ 的特解。

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。

辅助方程有重根 $6$，所以互补函数含 $te^{6t}$。特解同时试正弦和余弦，最后用两个零初值。

The auxiliary equation $(m-6)^2=0$ gives the complementary function $(A+Bt)e^{6t}$. For the particular integral, take $x_{\mathrm p}=p\sin t+q\cos t$. Then $x_{\mathrm p}'=p\cos t-q\sin t$ and $x_{\mathrm p}''=-p\sin t-q\cos t$. Substitution gives

Matching coefficients gives

The second equation gives $q=12p/35$. Substituting into the first gives $1369p/35=37$, hence $p=35/37$ and $q=12/37$. Thus the general solution and its derivative are

The initial conditions give

Thus $A=-12/37$ and $B=1$, so