求微分方程

满足当 $t=0$ 时，$x=7$ 且 $\dfrac{\mathrm{d}x}{\mathrm{d}t}=0$ 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 $-1/7$。互补函数要包含 $te^{-t/7}$，特解试作 $ke^{t/7}$，最后代入初始条件。

The auxiliary equation is $(7m+1)^2=0$, so the complementary function is $(A+Bt)e^{-t/7}$. For the trial $x_p=ke^{t/7}$, its derivatives are $x_p'=ke^{t/7}/7$ and $x_p''=ke^{t/7}/49$. Substituting gives $(k+2k+k)e^{t/7}=196e^{t/7}$, so $k=49$. The general solution and its derivative are

At $t=0$, $x(0)=A+49=7$ and $x'(0)=B-A/7+7=0$. Thus $A=-42$ and $B=-13$. Therefore